38

efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

  • Upload
    others

  • View
    0

  • Download
    0

Embed Size (px)

Citation preview

Page 1: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

����������� �������������������������� �!��"# ���%$

&('*),+.-0/214365�798;:07=<

>�?A@CBEDF?HGIBEJKD�LNMALPOQ?RG*S�TLVUW?RGXJZY,[\L2]_^�`!a�JKbRLVD_]_JXG;TLPcd?ED2Je]2fhg\[9ME^�ijDC],?Ek�^ml.Dn?Ea�o2L

>%L�opGX[0D�L2]�qHJKbRLVar?RGsG*S;LgoVS0BtBRum?EaMwvxBRaty,L,DnL,a�o_LWBEazcjDnBA@�?A@,JKu{JXGXkr|}S;LCBED2k

|9D2JIL2]2G~LV^s�R�RfF�A�WO4?Ek��0�A���

�\���H�t�A�,�,�����

���%���~�E���~�A� �x�F�,�,�2�E�!�d�C I�A� ¡p¢e£�¤Z¡E�H� ¥{�

Page 2: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8
Page 3: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

�������������

��������������������� � !�"�

! #%$�&'����(')*&+�,� -.��� (/�0�1�2���03��0��4657� -.89�:$�&;5�5�)*�68<��� !�"� =+>@? ACBEDGF H@I<J:KLB�MNH@OQP >R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU:==+>V= WYX�OZK[J�I]\E^_X�PS>S>S>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU:`=+>V` a1B�MNHb>S>G>R>S>S>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU�c=+>dc efI;g�XQMNPLXGhGJ�PLKLX�^_XQijIkD7J:KLMNHmln>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU�c

o #%$�&p8<���E�����k�q�����;3r&'st$1&j��� (u&+��� � !�"]v`+>@? w.X�H_\Ex;K[P >R>S>S>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU:y`+>V= AzhS{|K[x<XQB�M[X�D}>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=EU:~`+>V` ��^�J:K[P�>S>G>R>S>S>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~j?

� �%� ��-�� ��(u863C�0���j86����86���E& !6�� c]>@? ��J�P���H_MNJ�ORB���XQMNJ:KLB�M�>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~j?c]>V= �2���.J�I6�TK[x]X%�M[XQX�I���PY�0�<I<OZK[H_BEI�>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~�=c]>V` ��IkJ�M[X�OZK[J�I<\E^_X >R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~�`c]>dc ��BE�6I<�]X��k�]BED,J�H@I<P�>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~�cc]>Vy w.X�H_\Ex;K.D�BED7X�IqK�>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~��c]>V� aYx]X%��J��<PNP[H@J�IT��M[XQXR�<X�^@�9>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�~�~

v �Y� s&;���E� 3�8]���r���¡�����E���r�����E8 3�st3�86��8<�¢4 ��86s�$�5 !6�6£y+>@? ��6J�^@H_K�i >R>S>S>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�¤:¥y+>V= ¦¨§u©ªKLMNJ�I<PL��B�MND7J:K[H@BEI«>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�¤:¥y+>V` efPLB�M¬J��<H@J�^�X�D%F�X��6�<H@I]\EP�>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�¤�=y+>dc {�XQM¬H_B+�6H@O\�MNJ:�6x<P�>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�¤�`y+>Vy ­�J�H@IkK[x]XQB�M[X�D®>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T=�¤�`

¯ ������&+46��8<)��030�r�¬°|8 �1�����0(u&;��5�� �±�E�:�r�����E863zst3r8 �18<�¢46��8<st$15 !6£6¯�+>@? ��XQKLXQM¬D7H@I<J�I;K[P²>S>R>S>G>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T`:¥�¥�+>V= efPLB�M¬J��<H@J�^�X�D%F�X��6�<H@I]\EP�>R>S>S>S>R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T`:¥]?�+>V` w.i+��XQM[F�BE^@H@OH@�]X�J�^���BE^_ijx]X��]MNJ >R>S>G>R>S>S>R>S>S>G>R>S>S>R>S>S>G>R>T`:¥E=

³ &+-�&+��&j���E&;5 o ´6o

Page 4: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8
Page 5: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =EU+?

% & ��('��*)�+-,��(.N���

/ s*&+��-�&+���/(u8]���:$��0��4 B���J'\�MNJ:�6x|H�P�J P[�]F6PLXQK�B��X��]\�X�P"0.x<H@O¬x/OZB�g�XQMNP¢XQg�XQM[ig�XQM[KLXZl¢XZl]J�OZK[^_i,BEI6OZX21+K[x6J:KzH�P�1;��B�MCXQg�XQM[i,g�XQM[KLXZlnK[x]XQM[X�H@P�XZljJ�OZK[^_i¢BEI]XX��]\�XRH@I,K[x<XPLXQK-0.H_K[x/K[x<J:K�g�XQM[KLXZl J�P X�I<�]��BEH@I;KQ>9aYx]X ���0(u&;�/(2����&j3 H@P K[x]XTPLK[�<�<i2B��.K[x<XPLXQK�B�����XQM[��X�OZK�D,J:K[ONx<H@I<\EPB���J43���BEP[P[H_F6^@i H@I<�6I<H_KLX65�\�MNJ:� x�> aYx]X�D7BEPLK70CX�^@^98;:jI]B#0�IXZljJ�D�� ^_X�H@P<0.x]X�I,K[x]X\�MNJ:�6xnH@P7=?>@1+�rB�M!0.x6H@ONx,��XQMN�rX�OZKCD7J:K[ONx<H�I]\EPzJ:M[X.X�A+�<H_g:J�^_X�I;K3�g+H@J¢J¢PNH@D��6^_X%�<�<J�^@H_K iB5zKLB ��� (/�0�1�/���030�0�14 5 1�K[x<J:KRH@P�16K[H�^@H@I]\EP�B��K[x]X%� ^@J�I]X*0.H@K[x=DC9?RJ�I<�9?EC�= M[X�OZK[J�I]\E^_X�P�>

efISK[x]X*�<MNP[K�K[x]M[XQX*PLX�OZK[H_BEI6PF0CXPLK[�<�]i��]BED7H�I]BzK[H@^@H@I]\EP�B��jK[x]X � ^@J�I]X*J�I<�RB��j�6I6H_KLX��BE^@i;\�BEI<J�^tMNXQ\EH_BEI<P�1�B�MSX�A;�6H_g�J�^@X�IqK[^_i211��XQMN�rX�OZKGD,J:K[ONx<H@I<\EPRBEIG= > J�I<�2P[�]F<\�MNJ:� x<PB��H=?>Q>

efIGK[x]Xz^@J�PLK1K�0�B�PLX�OZK[H_BEI6PI0CXCP[K[�<�]i�K[x]X��thJ8 ��XQMNOZBE^�J:K[H_BEI%D7B+�]X�^;J�I<�SK[x<X��<H@D7XQMD�B+�<X�^�BEIkJ7D�B�M[XG\�X�I]XQMNJ�^��0J�D7H@^_i¢B���� ^@J�I<J:M�\�MNJ:�6x6PQ>

K LNM � �*+DOQP¢�7'²�SRG)��TOU.N� � �(.WV�.N�-X�� �SRZY[, M �*� �(Pn�SY?'\)

�tH_\E�]MNX7?^] �%�_�����2�  I�`_����ba��C¥��s¤;c,¡E���I�

/ ��J�D�BE�6P�M[X�P[�<^@K�B��]hSJ�P[KLX�^_XQi+Iedd~6f;J�I<�Sa�X�D���XQMN^_XQiJ�I<�S�tH@P[x<XQM<dm?�~bf;OZBE�<I;K[P�K[x<XI;�<D%F�XQMzB����]BED,H@I]B K[H@^@H@I]\EP�B��tJ O¬x]X�P[PLF�BEJ:MN�G3�B�M�J�IqinB�K[x]XQM�M[X�OZK[J�I]\E�6^@J:MYM[XQ\EH_BEIg5¬>

Page 6: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=EU:= ����� �����#���efITK[x6H@P�PLX�OZK[H_BEI 0CXSXZlj�6^@J�H@IThGJ�PLKLX�^_XQijI���P��<M[B+B��f>��� �� ������������� ������� ! XQK#" F�X%K[x]X�MNXQ\EH_BEI'F�BE�<I<�]X�� F;ipJTP[H@D�� ^_X%OQ^_BEP[X��'��BE^@i;\�BEI<J�^tOQ�]MNg�X H�I =?>Z> /�]BED7H@I]B¢K[H@^@H@I<\7B��$" OZB�MNM[X�PL��BEI<�<P�KLBnJn��XQM[��X�OZKD,J:K[ONx<H@I<\¢B���%-1�K[x<X �<�6J�^�\�MNJ:�6xB���"-]$%¨x<J�P�J g�XQM[KLXZln�rB�MzX�J�O¬xT^@J:KLK[H@OZX�P�A;�6J:M[XH�I�"-1 0.H_K[x¢K 0CB g�XQM[K[H@OZX�P�J��'&[J�OZX�IqKH_��J�I<�kBEI<^_i H_�1K[x]XSOZB�MNM[X�PL��BEI<�<H@I<\ ^�J:KLK[H@OZXGP�A+�<J:M[X�PYPNx<J:M[XSJ�ITX��]\�X�>

#%$�&;� ��&j( )(+* ?�]2G~L,u L,k�a�^=�-,�.R�0/(|}S;Lsa�[;U�@CL,DsBFy�MABEU�JKa�B(GXJKu{JKa0qE]sBFy " JK]�1 2 �]XQK6� 2 ^3 S;LVD�L � Je]�G*S9L 3 LVJ qES0G~L�MQ?HM54V?Ho2LVa�otk U6?HGXD_J764BFyNG*S;LWqHDF?98\S % ^ 3 JXG*SQS0BED2J7:pBEa\GI?EuL�Mtq0L2] 3 LVJ qES0G~L�M ? ?Ea�M bEL,D2GXJZoC?EuxL�Mtq0L2] 3 L,J qRS G~L�M<;=?> §S?-@

�]B�M.XZl]J�D��6^@XR�rB�MYK[x<X%=DC�` M[XQ\EH@BEIkH@IT�tH_\E�]M[XG= 16K[x]XGD7J:KLMNH_l � H@P

� =ABBBBBBC¥ ¥ ¥ ; ?«¥¥ ¥ ¥ ? ; ?¥ ¥ ¥ ¥ ? ;; ? ¥ ¥ ¥ ¥? ; ? ¥ ¥ ¥¥ ? ; ¥ ¥ ¥

DFEEEEEEGIH0.x]BEPLXS�]XQKLXQM¬D7H@I<J�I;KYx<J�PYJ:F6P[BE^@�]KLXRg:J�^@�]XG¤ = `^>:>

2

3

6

i i i1

1 1

5

1

4

1

�tH_\E�]M[XG= ]�JLKNM �I�F�h ~�_�baW_����C�E����¡E��_ aC�~�_£�� �O ����� -QP R H@I<OZX%K[x]X \�MNJ:�6xS% H�P.F6H@�6J:M[K[H_KLX 3�BEI<X OQJ�I'OZBE^_B�Mg�XQM[K[H@OZX�P�F ^@J�O�:kJ�I<�0.x<H_KLX�P[BuK[x<J:KnF6^@J�O�: g�XQM[K[H@OZX�PTJ:M[X BEI<^_i|J��'&[J�OZX�I;K¢KLB 0.x<H@KLXkg�XQM[K[H@OZX�P J�I<�±g+H�OZXg�XQMNP[J�5z� OQJ�IkF�X�0YM¬H_KLKLX�IT� =UT ¥ VVXW}¥IY 16PLBD0CXGD �<PLK�XQg:J�^@�<J:KLX

�]XQKZV =\[]�^�_a` PL\EI�b�c � ]�d ��egfFfFf c�h ]�d h e

Page 7: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =EU:`

� J�ONx'KLXQM¬D H@I�K[x6H@P�P[�<DbH@P��QXQM[BT�<I<^_X�P[P�F6^@J�O�:�g�XQM[KLXZl b 3 ; 5H@P�J��'&[J�OZX�IqK�KLB 0.x6H_KLXg�XQM[KLXZl ; �rB�MRX�J�O¬x ; > aYx]XQM[XQ��B�M[X-0CX x6J�g�X7XZl]J�OZK[^_ipBEI]X�KLXQMND ��B�MRX�J�ONx2D,J:K[ONx<H@I<\B�� % >�e KYPN���,OZX�P�KLB,PNx]B#0 K[x<J:KYK 0CB,I]BEI��QXQM[B7KLXQMND7P.x<J�g�X%K[x]XSP[J�D�XGPNH_\EI�>R �]�<��BEPLX40CXkx<J�g�X K 0CBuD7J:K[O¬x<H@I]\EP��Y�]MNJ�0 K[x]X�D BEI]XTBEI¡KLB��|B���K[x]XTB�K[x]XQM�>aYx]X�H_MRP[�]��XQM[��BEP[H@K[H_BEI�H@P�JT�<I6H_BEI�B���30�]BE�]F ^_X�� 5YX��]\�X�PSJ�I<� �<H�P & BEH�IqKOZi+OQ^_X�P�>���I<XOQJ�I¢KLM¬J�I<PL��B�MND K[x<X��<M¬PLK�D7J:K[O¬x<H@I]\ H@I;KLB%K[x<XPLX�OZBEI6�¢F+i� M[B�K[J:K[H�I]\� X�J�ONxTOZijOQ^_XH�IK[�]MNI�>te I7�6J:M[K[H�OQ�<^@J:M�H@KP[���,OZX�P�KLB P[x]B#0 K[x<J:KCH_��K 0CB D7J:K[ONx<H�I]\EP��6H ��XQM�BEI<^_i7J�^_BEI<\JpPNH@I]\E^_X,OZijOQ^_X21*K[x<X OZB�MNM[X�PL��BEI<�<H@I<\kKLXQMND7P H@IuK[x]X �]XQKLXQMND7H@I6J�IqK x<J�g�X K[x]X P[J�D�XP[H_\EI�>

aYx<H@P H�P*P[�]�<� ^@H_X��%F;i K[x]X���BE^@^_B60.H@I]\S^_X�D,D7J 1�0.x<H�ONx�H@P X�J�PNH@^_i �<M[B�g�X��,F;i H@I<�6�<O8K[H_BEI�> �� &j(u(u8� >%L,G�� =������ H fFfFf H � >�� ��� H � >�� =������ @CLw?Q]_J*UX8u L otu B�]tL�M o,[;D2bEL JKa=?> @ >%L,G�� � @CL G*S;L�8DFBtMR[9opGsBnyPG*S9L 3 L,J qRS G*] BEa LCMtq LV]������ � H � > �� H fFfFf ?Ea�M!� > G*S;L8�DFBpME[�otG BFyWG*S;L 3 LVJ qES0G*] BEa G*S9LWD�LVUW?EJ*a�J*a qQL�Mpq L2]"� � � > H fFfFf$# DnLCoC?Eu*u�G*S0?HG bRLVDVGXJXo_?EuL�Mtq0L2] ?ED�L 3 LVJ qES G~LCM ; ?Ea�M4S0BED2J7:pBEa\GI?Eu LCMtq LV] ?&% @ |}S;L,a'� � = 3 §S?#5 h�( � ( � � > ^3 S;LVD�L") JK]�G*S9L6a�[;U�@CL,DWBFy�bRLVDVGXJXo2L2]�]2GXD2JXopGXu k L,a�o,u B�]tL�M @Vk+* ?Ea�M c Je] ?�,:= BFy�G*S;Lu L,a qHG*SzBFy-* @�� � .$/g� � � �10324/ R �]�<��BEPLX%K[x]X \�MNJ:�6x % H@P�J�I � F+i ) M[X�OZK[J�I]\E^_X21I0.H_K[x g�XQM[K[H@OZX�P � ? H = H fFfFf H �5� C� ? H = H fFfFf H )6� >a�B�OZBED��6�]KLXRK[x<X��<XQKLXQMND7H@I<J�I;KCB���H_K[PzJ��'&[J�OZX�I<OZi D7J:KLMNHml(1 0�XROZBED-8�6�]KLXSH_K[P�X�H@\�X�Iqg�X�OZKLB�MNPJ�I<�TK[x<X�H_M�X�H_\�X�I;g�J�^@�<X�PQ>

�]B�M�7 H c98 = �<XQ�6I]X

: 3<; H = 5 = P[H�I?> 7�;�A@ ? P[H@IB> c =)C@ ? fe KYH@PYX�J�PLi KLB,ONx]X�O�:TK[x<J:K : H@PYJ�IkX�H_\�X�Iqg�X�OZKLB�M�B��t� 1B0.H_K[xTX�H_\�X�I;g:J�^@�]X

=tOZBEP > 7�A@ ? @ = ; OZBEP > c)D@ ? fE Xp�<H��<I��dK �6�<^@^zK[x6H@P���B�MND%�6^@J BE�]K¢B���Jux<J:K�]'P[H@I<OZX % H@P7JF� �<M[Bj�<�<OZKG B��RK�0�B^@H@I]XS\�M¬J:�6x<P�1 K[x]X%X�H@\�X�Iqg�X�OZKLB�MNPRJ:MNX%K[x]X �<M[Bj�<�<OZK.B�� K[x]X%X�H@\�X�Iqg�X�OZKLB�MNP�B��K[x]X ^�H@I]X\�MNJ:�6x<PYH�I<�<H_gjH@�<�<J�^�^_i�>

/ PH7 H c g:J:M[i B�g�XQMzH@I;KLXQ\�XQMNPH@I dm? H � f�C dm? H ) f;1+K[x<X : �rB�M¬D J�I�B�M[K[x]B�\�BEI6J�^<F6J�P[H�PB���X�H_\�X�I;g�X�OZKLB�MNPQ>zaYx]XQMNXQ�rB�M[XRK[x<XG�]XQKLXQMND7H�I<J�IqK�B��t� H@P

�]XQK<� =JIKL&M �

hK� M �

=tOZBEP > 7�A@ ? @ = ; OZBEP > c)D@ ? f

Page 8: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=EU�c ����� �����#���� g:J�^@�<J:K[H@I<\ 1 2 �]XQK<� 2 ��B�M �\= ) = ~T\EH@g�X�P¢?�=�¤�~�~�~j?��pK[H@^�H@I]\EP�B��YJ�O¬x]X�P[P 8

F�BEJ:M¬��>��I]X*D7J�iRP[x]B#02K[x<J:K���B�M � H ) F�B�K[xR^@J:M[\�X*K[x]X*I+�<D%F�XQM�B��+K[H�^@H@I]\EP�H@P�XZlj� 3 % � ) , > @� 3 �A@') 5�5 H 0�x]XQM[X<% = ?�§ � �� @ �� � § �� � @ fFfFf H@P�AzJ:K[J�^@J�I���PYOZBEI<PLK[J�I;KQ>���� � ����

e KJ0.H@^�^�F�X �6PLXQ���6^�P[x<B�M[K[^_ikKLBTOZBED��6�<KLX K[x<X I+�<D%F�XQMB��K[H@^@H@I]\EP���B�MSJ¢\�M¬J:�6xpBEIpJKLB�MN�<P.J�P 0CX�^@^<3�X�>d\]>J7M[X�OZK[J�I]\E^@XE0.H_K[x H_K[PYB��6��BEPNH_KLXGP[H@�<X�P.H@�]X�I;K[H_�<X��g5¬>ChSJ�P[KLX�^_XQi+IP[x]B#0CX���K[x<J:KTK[x6H@P OZBE�<^@��F�X'J�OQOZBED7�6^@H@P[x<X��±F+i J/^�H@I]X�J:MTOZBEDGF H@I<J:K[H_BEI B�� �rBE�<M�]XQKLXQMND7H@I6J�IqK[PQ>�e I|X�PNPLX�I<OZX�K[x<X�PLX ��BE�]Mn�]XQKLXQMND7H@I6J�IqK[P¢OZB�M[M[X�PL��BEI<�¡KLBuK[x]X �rBE�<MH@I]X�A+�<H_g:J�^_X�IqK �L�<H@P[OZM[XQKLX'PL�6H�I¡PLKLMN�6OZK[�]M[X�PGuBEI¡K[x<Xp\�M¬J:�6x)% X�D%F�X��<�]X��±BEI±K[x<XKLB�MN�<PQ> W.J:K[x<XQMK[x<J�I,\�B%H@I;KLB%�]XQK[J�H�^@P�1�0CX &[�<PLK*I]B�KLX�K[x<J:KK[x<X.J:F�B�g�X�<M[B+B�� �0J�H@^@P BEIJ�KLB�M¬�<PtP[H@I<OZXCK 0CBSD7J:K[ONx6H@I]\EPtD7J�i �<H ��XQM�BEI J�^_B+B��-0.x<H@O¬xG\�B+X�P J:M[BE�<I<� K[x]XCKLB�MN�<P30H@PRI]B�KSI+�<^@^98�x]BED7BE^_B�\�BE�<PW5¬> aYx]X7O¬x<J�I]\�X,H@I'P[H@\EI H@I'K[x]X7�]XQKLXQMND,H@I<J�I;K��M[BED BEI<XD7J:K[ONx<H�I]\�KLBK[x<XzI<XZl+KtH@P�K[x]X�I J���<I6OZK[H_BEIGB�� K[x]X�x]BED�BE^_B�\�i OQ^@J�P[PtH@I � 3�� H =-,:=^=S5B���K[x<XS^_B;B��1>aYx]XRM[X�PN�<^_K�H@P�K[x<J:K 3��rB�M � H ) F�B�K[xTXQg�X�Ig5zK[x]XSI+�<D%F�XQM.B���K[H@^�H@I]\EPzB��J�I � C ) KLB�MN�<P.H@P

I��h = ?

= 3 §�� ���1@ � � � @ � � �1@ � �L� 5 H0.x]XQM[X� ]�� = K

��� M d ����e�� K� ` M d ����e�� T���@ ?

� @ ;�� @ ;� Y f

� B�KLX%K[x<J:K.J�OZK[�<J�^@^@i�� ��� = ¥7H@I K[x6H@P�OQJ�PLX�>��<B�M��]XQK[J�H@^�P�PLXQX dd~bf�B�M-d c2f >��I]XSOQJ�I P[x]B#0 K[x<J:K

^�H@DI �h"!$# ?

� ) ^_B�\ I �h = %

> f���% & �('3/Z�� /*) � ��/ 2 /,+��I� �������.-E XSJ:M[X��<MNH@D7J:M¬H@^_i H@IqKLXQMNX�PLKLX��TH@I¢K[x]X ������-��6�:( (u&;8<5��1��& BEIn��XQM[�rX�OZK�D7J:K[O¬x<H@I]\EPB���J�\�M¬J:�6x % >

/ g�XQM[in�6PLXQ���6^�OZBEI<PLX�A+�]X�I<OZXSB�� hSJ�PLKLX�^@XQi+I���P.OZBE�<IqK[H�I]\�K[x]XQB�M[X�D H@P

#%$�&;� ��&j( !0/21r ;´4365 >}L,G87 = � 3 � � H 9 � 5 H fFfFf H 3 � � H 9 � 5 � @CL ?z]2[�@V],LtG BFy MEJ*UWL,DC]_^# 3 JXG*S G*S;L 7 G*S MEJ*U�LVD o_BEbEL,D_J*a q 3 SAJXG~LQbRLVDVG~LQ6:� L ?Ra�M @,u ?Ao<; bEL,D2G~L 6 9 L %�@ |}S;L8�DFB @�?A@,JKu{J*GXk G*S0?HGj?RuKu%MRJKU�LVD_](J*a=7 ? 8-8�L�?ED�JKaQ? [;a�J yVBED2U�u k oVS;B�],L,awUW?RGIo,SAJ*a q JK]{M#3 7 5 = 2 �]XQK#30� ��� 3 �?> H 9 L 5�5 �A@ > � L @ � 2 f

Page 9: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =EU:y

aYx]X.J��]g:J�I;K[J:\�X�B���K[x<H@P*M[X�P[�<^_K*H�P K[x<J:KK[x<X.�]XQKLXQMND7H�I<J�IqK�rB�M�K[x]X&LBEH@IqK�<M[B�F6J:F6H�^@H_K�iB���cG�<H@D�XQM¬P�H@P�B��]P[H �QX c J*a�MAL�8�LVa�MAL,a\GXu{k BFysG*S9Ls]_J :pLdBFy�G*S;L�UW?RGXD_J76 �u>�efIG�6J:M[K[H@OQ�6^@J:MKLB�OZBED7�6�]KLXzK[x]XC�<M[B�F6J:F6H�^@H_K�i�K[x<J:K J�P[H@I<\E^_XX��]\�X�J:�<��X�J:MNP�1�BEI]X &L�<P[K�I]XQX��<PtJ�P[H@I<\E^_XX�^_X�D�X�I;KYB���� ��� >

e � 0CX%OQJ�IpOZBED��6�<KLX J�PLijD��<KLB�K[H�OQP.B��*�u���.��B�M^�J:M[\�XGM[X�OZK[J�I<\E^_X�PS0CX OQJ�I�K[x]X�I�<I<�]XQMNP[K[J�I<� X�J�P[H@^_i%K[x]X.^@H�D7H_K[H@I]\�D�X�J�P[�<M[X�P BEI7K[H�^@H@I]\EP�B���K[x]XY�6^�J�I]X�> E X���^�^<�]BSK[x<H�P^@J:KLXQM�>

/ D�B�MNXR\�X�I]XQMNJ�^�OQ^@J�P[PzB���D�X�J�P[�]MNX�PzBEIT��XQM[�rX�OZK�D7J:K[O¬x<H@I]\EP��6PLX�Pz\�MNJ:�6x<P�0.H@K[x0�X�H_\Ex;KLX���X��]\�X�PQ> ef�RX��]\�X�P J:M[X 0�X�H_\Ex;KLX����F3��65�� ¥ 1YK[x]X 0CX�H_\Ex;K¢B��SJu��XQM[��X�OZKD7J:K[ONx<H�I]\ H�PK[x]X�<M[Bj�<�<OZK�B��1H_K[PCX��]\�XT0�X�H@\ExqK[P�1]J�I<�nK[x]X�<M[B�F J:F6H@^@H_K i�D�X�J�P[�]M[X 0�XJ:M[XYH@I;KLXQM[X�PLKLX��,H�I H�PtK[x<J:K 0.x<H�ONx \EH_g�X�P*JGD7J:K[ONx<H�I]\SJR�<M[B�F J:F6H@^@H_K iR�<M[B���B�M[K[H_BEI6J�^;KLBH_K[PS0CX�H_\Ex;K-3���B�M�6I<H_KLX%\�MNJ:�6x<PW5¬>�aYx]XQM[X [email protected][H@D�� ^_XS\�X�I]XQMNJ�^�H ��J:K[H_BEI B��*aYx]XQB�M[X�D7P?SJ�I<�k= ��B�M�0CX�H_\ExqKLX���\�MNJ:�6x<P�>

E X P[x<BE�<^@��J�^@PLB ��BEH@I;K BE�]K7K[x<J:K7g�XQM¬P[H_BEI<P B��RaYx]XQB�MNX�D7PT?kJ�I<�¡='x<BE^@���rB�MJ:M[F6H_KLMNJ:MNi,�6^@J�I6J:M�\�MNJ:�6x<P�1<P[XQX dd¤bf1J�I<�Zdm?Q¥2f >

� LNM � Y?'\,��(.�, , .�'\, VN��� M �� � O �� ���

aYx]XB�MN�]XQM�8 ) / �QKLX�OR�6H@J�D�BEI<�nH@P�K[x]X�M[XQ\EH_BEI P[x]B60.I�>te K!0zJ�P��]XQ�6I]X��nJ�I<� PLK[�<�6H_X��H@I dd� 1j`6f > / M¬J�I<�]BED K[H�^@H@I]\SH@PJ�^@PLB%PNx]B#0.I�> E x;i7�]B;X�P�H_KI]B�KC^_B+B2: x<BED�B�\�X�I]XQBE�<P�1J�P.�]B+X�P�K[x]XRK[H@^�H@I]\ B���J,P�A;�6J:M[X

� X�J:MtK[x]XzOZB�MNI]XQM¬P�1:i�BE�%BEI6^_iRPLXQX�K[H�^_X�P1B��]BEI<X�:K i;��X�j> aYx]X<0CJ�iGKLB��<I<�]XQMNP[K[J�I<�K[x<H@Pz�6x]X�I<BED�X�I]BEIkH@PzgjH@J K[x]X%x]X�H_\Ex;K����<I6OZK[H_BEI�>

��� �/ � 0 � �- / �]BED,H@I]B K[H@^@H�I]\/OQJ�I F�X K[x]BE�]\Ex;KTB�� J�P J�I H@I;KLXQM[�0J�OZX2H@I = @ ? �6H@D�X�I<P[H@BEI<PQ>R ��X�OQH_�6OQJ�^@^_i21�KLB|J�K[H@^@H@I<\ 0CX2J�P[PLBjOQH@J:KLX2J�I�H�IqKLXQ\�XQM�8 g:J�^@�]X�� ���<I6OZK[H_BEI���BEI�K[x<Xg�XQM[K[H@OZX�P B��.K[x<XnMNXQ\EH_BEI/K[H@^_X��\1*J�P ��BE^@^_B60.PQ>9W.J:K[x<XQM K[x6J�I �]XQ�6I]XnK[x]XTx]X�H_\Ex;K�1 0�X�]XQ�6I]XSK[x<XGx]X�H_\Ex;K.�<H ��XQM[X�I<OZXGF�XQK�0�XQX�I K 0CB J��'&LJ�OZX�I;K�^@J:KLK[H@OZXS��BEH@I;K[PQ>Ce �tX��]\�X � �H@PzI]B�KYOZM[BEP[PLX�� F+i¢J��<BED7H@I]BB1+K[x]XSx]X�H_\Ex;K��<H ��XQM[X�I<OZX��I3 � 5�§�� 3 � 5CH@PR?�H_�1K[x]XR��J�OZXKLB9K[x]Xk^_XQ��K B�� � � H@P F ^@J�O�:g1CJ�I6��§S? B�K[x]XQM�0.H�PLX�>�e �K[x]XTX��]\�X�H�P�OZM[BEP[PLX�� F+iuJ�]BED7H@I]BB1��I3 � 5�§�� 3 � 5�H@P§.`SB�M @ `%J�OQOZB�MN�<H�I]\%KLB 0.x]XQK[x<XQMCK[x<X��0J�OZX�KLB%K[x]X�^_XQ��KCH�PF6^@J�O�:�B�M!0.x6H_KLX�> aYx<H@P*�<XQ�6I]X�PCJ � BEI]X@8 ��B�MND 1jK[x<J:KzH@PCJG�0�<I<OZK[H@BEI7BEI¢K[x]X�B�M¬H_X�IqKLX��X��]\�X�PRB��zK[x]X�\�MNJ:�6x % P[J:K[H@PL��ijH@I]\ : 3 §��65 = § : 3��65¬>7­�B�M[XQB�g�XQM K[x]X7BEI]X@8 ��B�MND H�POQ^_BEPLX��\] K[x]XkP[�6D J:M[BE�<I<� J�I;iuB�M¬H_X�IqKLX��¡OZi+OQ^_XkH�P �QXQM[B 30P[H@I6OZXnK[x<XkP[�<D J:M[BE�<I6�J�Iqin��J�OZX%H@P �QXQM[BB]J7�<BED7H@I]B OZM[BEP[PLX�P.XZljJ�OZK[^@i¢BEI]XRX��<\�XSB���X�J�O¬xk�0J�OZX65¬>�aYx<XQM[XQ�rB�MNX

Page 10: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=EU:� ����� �����#���K[x]XQM[XTH@P�Jp���6I<OZK[H_BEI �\1<0�X�^@^��<XQ�6I]X����]��KLB9J�I J��<�<H_K[H@g�XnOZBEI<P[K[J�IqK�1�0.H_K[x/K[x]X�PLXX��]\�X@8��<H ��XQM[X�I<OZX�PQ>

��H@\E�]M[XG` ] �!�C�E�p�2�  I� _����ba��_¥ ���  I�F� �p� �_���2���

aYx]Xnx]X�H_\Ex;KG�0�<I<OZK[H_BEI2B��.J�K[H�^@H@I]\T\EH_g�X�P JpD7J:�/�rM[BED =?>GKLB =�> �RI]X,B��YK[x<X�<MNH@I<OQH@�6J�^]D�B�K[H_g:J:K[H_BEI<PC��B�MCPLK[�6�]i+H�I]\G�]BED7H�I]BEP�H�PKLB PLK[�<�]i�K[x<XD�Bj�]X�^ B���MNJ�I<�<BEDD7J:�6P�B���K[x<H@PYP[B�M[KQ>

� B�KLX�K[x<J:K�JSK[H@^@H@I]\R�]XQKLXQM¬D7H@I]X�P*JGx]X�H_\Ex;K ���<I6OZK[H_BEI\1qJ�I<�[email protected]�XQMNP[Jj> / ^@PLBB1qK[x<Xx]X�H_\Ex;K.�0�<I<OZK[H_BEI�BEI�K[x]X%F�BE�<I<�<J:M[inB�� K[x]X%M[XQ\EH_BEI KLB¢F�XSK[H@^@X���H@P�H@I<�]XQ��X�I<�]X�I;K�B��K[x]XSK[H@^@H�I]\]>

�]B�M.K[x<X%= ) Ck= ) P�A+�<J:M[X21]K[x]XRF�BE�<I6�<J:M[i,�0�<I<OZK[H@BEITH@P�J�PLijD��<KLB�K[H@OQJ�^@^@i��6J:K�16H�I��J�OZKJ�P[P[�<D7H�I]\ K[x]XSg:J�^@�]XSJ:K.J7OZB�M¬I]XQM�H@P.¥ 16H_K.J�^_KLXQM¬I<J:KLX�PYBEITX��<\�XGX��]\�XSF�XQK�0�XQX�I¥ J�I6� ?�B�MY¥ J�I<� §G?^]*PLXQXR��H@\E�]M[X�c]>*w.B60�XQg�XQM��rB�M�K[x<X / �QKLX�OS�<H�J�D�BEI<�¢B��1B�MN�]XQM) 1<H_KYH@P�^@H@I<X�J:M�BEITX�J�ONx X��]\�X2]

�]B�M JkI]BEI 8 �6^�J�I<J:MSM[XQ\EH_BEI2P[�<ONxuJ�PGJ�KLB�MN�<P�1�K[x]X,x]X�H_\Ex;KGD7H@\ExqKGI<B�KSF�X�0�X�^�^98�]XQ�6I]X��\]¢H_K OQJ�I�F�Xe0�X�^�^98��]XQ�6I]X��u^_BjOQJ�^@^_i21 F6�]K BEI�Jp�6J:K[x 0.H@I<�6H@I]\kJ:M[BE�<I6�uK[x<X

Page 11: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =EU�U

3

1 0

2

0

0 1 0 1 0

-1

-1-1-2 -2

2

1 0

-1

-1

�tH_\E�]M[XRcB]�� �F� a2�, I¤=��� �j�p�2�������  I� _ ���6a��

0

0

32

11

23

4 5

6 7

7

8

7

7

6

6

5

5 5

6

44

4

3 322

1 1

8

��H@\E�]M[XGy ]�� �F� a_�t I¤��_���R�2¡��E�����Z� �_¥��C� ���  I���x��� �_���_�E�H�

Page 12: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=EU:~ ����� �����#���KLB�MN�<PtH_K�D,J�i%x6J�g�X�J�I<BEI 8 KLMNH_gjH@J�^E��XQMNH@B+��>1efI<�<XQX��\1:�rB�M J�\�MNJ:�6x%BEI%K[x<X � C ) KLB�MN�<PK[x]X x<X�H_\ExqK 0.H@^@^1x<J�g�X�J 0�X�^@^�8��]XQ�6I]X��kx<B�MNH �QBEI;K[J�^���XQMNH@B+�����¢J�I<� g�XQM[K[H@OQJ�^���XQMNH_Bj�� � 1�F�B�K[x H@I;KLXQ\�XQMNP�1�PN�<ONx'K[x<J:KRK[x<X7x]X�H_\Ex;K�O¬x<J�I]\�X�BEI J�I;ipOQ^@BEPLX��'�6J:K[x 0.H@I6�<H@I]\BEI<OZX2x]B�MNH �QBEIqK[J�^@^@i±J:M[BE�<I6� K[x]X9KLB�MN�<P�H@P � � 1RJ�I<��P[H�D7H@^@J:M[i � � ��B�Mkg�XQM[K[H�OQJ�^@^_i 80.H@I<�<H�I]\G�6J:K[x6PQ>

aYx]X7x]X�H@\ExqKRB��zJTMNJ�I<�<BED K[H@^@H@I]\nBEI9JT� ^@J�I<J:M�M[XQ\EH_BEI9H�PRJTMNJ�I<�]BED �0�<I<OZK[[email protected]_K[xkK[x<BEPLXG\EH_g�X�I F�BE�<I<�<J:MNi¢g:J�^@�]X�PQ>.e KH@PYX�J�PLiTKLBn\E�]X�P[P�K[x<J:KK[x]X%x<X�H_\ExqKJ:KJ�IH@IqKLXQM¬H_B�M���BEH@I;KB��JnMNJ�I<�]BEDbK[H@^@H�I]\,B��J P�A+�<J:M[X B���PNH@�]X ) H@P��B3 ) 5¬>Gw�B#0CXQg�XQMS�rB�MK[x]X / �QKLX�O �<H@J�D�BEI6� H_KH@PI]B�KJ:KRJ�^@^�OQ^_X�J:MJ0.x<J:KK[x]X x]X�H_\Ex;K�B��J¢MNJ�I<�<BED K[H@^�H@I]\0.H@^@^CF�X�>|e I���J�OZKnH_KD0.H@^@^CK i;�6H�OQJ�^@^_iu^�H_XkI]X�J:M 3�0.H@K[x<H@I��B3 ) 5�5 B��RH_K[P7D�X�J�I|g�J�^@�<X�>­kB�M[XQB�g�XQMH@K[PYD�X�J�ITg:J�^@�]XSOQJ�ITF�XS�]XQKLXQM¬D7H@I]X��nF+i P[BE^_g+H�I]\ J,OZXQM[K[J�H@ITg:J:MNH@J:K[H_BEI6J�^�<MNH@I<OQH@�6^_X 30D,H@I<H@D7H ��H@I]\%J,OZXQM[K[J�H@ITX�I]XQMN\�iB5¬>a.x<J:K.H@PzK[x]X%OZBEIqKLX�I;K.B���K[x<H@PYP[X�OZK[H_BEI�>

�� � ��)�� � � / ���/ �aYx]X K[x]XQB�M[X�DbB��CACBEx<IB8�hRX�Iqi�BEIB8�{*M[B��<� PLK[J:KLX�P�K[x6H@P�H�I�\�X�I]XQM¬J�^�]Y\EH_g�X�I'J��B�MN�6J�I�]BED7J�H@I� 0.H@K[xu�0�<I<OZK[H@BEI � ]����� �!1�J�I<� J PLX�A+�]X�I<OZX B����]BED7J�H@I<P H@I���=?>OZBEIqg�XQM[\EH�I]\�KLB� 1 0.H_K[xnF�BE�<I<�<J:MNi x<X�H_\ExqKz�0�<I<OZK[H_BEI6P ���OZBEI;g�XQM[\EH@I]\�KLB��I1]K[x6J:KzH�P� ������� 1�K[x]X�I-0.H_K[x%�<M[B�F6J:F H@^@H_K iKLX�I6�<H@I]\�KLB ?CK[x]X�x]X�H@\ExqK�B�� J�MNJ�I<�]BED¨K[H@^�H@I]\S0�H@^@^^@H_X I]X�J:M�3�0.H_K[x<H@I��B3 ) 5�5�H_K[P�D�X�J�I'g�J�^��]X21�J�I<�'K[x]X�D�X�J�Ipg�J�^��]X � � H@PK[x]X��<I<H�A+�]X���<I6OZK[H_BEI : D7J�l]H@D7H ��H@I<\ J7OZXQMNK[J�H@I � X�I;KLM[B��;i

� I;K#3 : 5 =���� X�I;K63�� � : H � � : 5! ;" = fw.XQM[XX�IqK#3�# H%$ 5 =

?

>3�& 3 >('*) 5 @ &T3 >('"+ 5 @ & 3 >('*, 5 @ & 3 >('"- 5�5 H0.x]XQM[X.&T3<;(5 = §0/ �� ^_B�\C=tPNH@I$ $H@P ! B�F6J�ONx]XQgjP�:+i ��P �0�<I<OZK[H_BEIF1�J�I<� '") H '"+ H '", H '"-J:M[XGOZXQM[K[J�H@Ik�<M[B�F6J:F6H�^@H_K[H_X�P<0.x<H@O¬xTJ:M[XG�]XQKLXQM¬D7H@I]X�� F+i¢K[x<XS�rBE^�^_B#0.H�I]\ X�A;�<J:K[H@BEI<PQ>

=B3 '") § '*+ 5 = #=B3 '"- § '*, 5 =

$' ) @ '"+ @ ' , @ '*- = ?PNH@I >('") P[H�I >('"+ = P[H@I >('", PNH@I >('"- f

aYx]X��<MNB;B���B��CK[x<X��<MNPLK��6J:M[KRB��zK[x<H�P�K[x]XQB�M[X�D H@PGPLBED�X�0.x<J:KSP[K[J�I<�<J:MN��>�aYx<Xx<J:MN�]XQM��6J:MNK.H@PzK[x]XGOZBED7�6�]K[J:K[H_BEITB���K[x]XS�0�<I<OZK[H@BEI X�IqK�>

aYx]X'H@�]X�JuB��GK[x]X'�<MNB;B��SB��%K[x]X'�<MNPLKn�6J:M[K H@PnK[x]X'��BE^@^_B#0�H@I]\]> � J�O¬x�x]X�H_\Ex;K���<I6OZK[H_BEI H@P�J ! H_� P[ONx<H@K����0�<I<OZK[H_BEIe0.H@K[x ! H_�6P[O¬x<H_K��ROZBEI<PLK[J�I;K.` 16P[H�I<OZX 2 : 3<; � H = � 51§

Page 13: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =EU:¤

: 3<; � @ ? H = � 5 2�� `J�I<� P[H@D,H@^@J:MN^_i���B�M K[x]X = 8��6H_M[X�OZK[H_BEI�>�efI �0J�OZK 0�X�x<J�g�XYJRPLKLM[BEI]\�XQMOZBEI<�<H_K[H_BEI K[x6J:K�0.x]X�I ; � § ; > H = � § = > J:M[XSXQg�X�IF12 : 3<; � H = � 5�§ : 3<; > H = > 5 2�� =tD7J�l � 2 ; � § ; > 2 H 2 = � § = > 2 � 3 ?#5! XQK & F�X7K[x]X,OQ^�J�P[PGB��z�0�<I<OZK[H_BEI6PSP[J:K[H@PL��ijH@I]\Z3 ?#5¬> E x]X�IG0CX¢P[OQJ�^@X,K[x]X ;�§ J�I<�= 8�OZB+B�MN�<H@I6J:KLX�P F;i���1�K[x]X�I � : H@P�PLK[H@^�^�H@I�&.> � B�KLXTK[x<J:K & H@P OZBED��6J�OZK 30H@I�K[x<X�<I<H_��B�MND KLB���BE^_B�\�iB5¬>

��InK[x<H@PzPL�6J�OZX21]K[x<X�0�<I<OZK[H@BEI<J�^ � I;KzH�PC�<�<��XQMCPLX�D,H@OZBEIqK[H�I;�]BE�6P�] H_K[PCg�J�^@�<X�OQJ�IBEI<^_iTH@I<OZM[X�J�P[X J:K�J¢^�H@D7H_K���BEH@I;KQ>YaYx<H@P.�rBE^�^_B#0.P���M[BED K[x]X PLKLMNH�OZK�OZBEI<OQJ�g+H_K i�B��*K[x<X���<I6OZK[H_BEI7X�I;K63�# H%$ 5¬>�ACBEI6OQJ�gjH_K�i,J�^�PLBG�<MNB�g�X�Pz�<I<H@OQH@K�i B���K[x]X�D7J�ljH�D7H ��H@I]\R�0�<I<OZK[H@BEI�>R H@I<OZX�& H@P�OZBED�� J�OZK�1�OZB�g�XQM%H_KJ0.H_K[x'Jn�6I<H_KLX I;�6DGF�XQM�B���D�XQKLM¬H@O F6J�^@^@P����B3��b5B��M¬J��<H@�<P��q> E X 0.H�^@^zP[x]B#0 K[x<J:K7��B�M,J�Iqi : 8 &�1CK[x]XkI+�<D%F�XQM�B���K[H@^�H@I]\EP �rB�M0.x<H@O¬x � �p^@H_X�P�0.H_K[x<H@I� B�� : H@P�XZlj� 3 �� � � I;K#3 : 5@3 ? @ �B3 ?#5�5�5¬>.aYx6H@PY�]XQKLXQMND,H@I]X�PYK[x<XP[H �QX B��*X�J�ONx��� 3��b5¬> aYx]XQM[XQ��B�M[X-0.x]X�I �RH@P�P[D7J�^�^�1 K[x<X I+�<D%F�XQMB��K[H@^@H@I]\EP 0.x]BEP[XI]B�MND7J�^@H �QX��Tx]X�H_\Ex;K����<I6OZK[H_BEIT^@H_X�P�OQ^_BEP[XSKLB : ����� B�g�XQM�0.x]X�^@D7H�I]\E^_i¢�<BED7H@I<J:KLX�P�K[x<XI;�<D%F�XQM�B��tB�K[x]XQMYK[H@^@H�I]\EP�1jXQg�X�I�OZBED%F6H@I]X���>

aYx+�<P�J�^@D�BEPLK�J�^@^�K[H@^@H@I<\EPC^�H_XSI]X�J:M : ����� >a1BpOZBE�<I;KSK[x]X¢I+�<D%F�XQMSB��zK[H@^@H@I]\EPJ0�x]BEPLX,x]X�H@\ExqKS�0�<I<OZK[H@BEI9^@H_X�PRI]X�J:M Jk\EH_g�X�I

���<I6OZK[H_BEI : 8 &S1�KLMNH@J�I]\E�<^�J:KLX. 0.H@K[x J¢�6I]X D�X�P[x\1�^@J:MN\�X%BEIpK[x]X�P[OQJ�^_X B�� �F �]K0.H_K[x'D�X�P[x9PNH �QX KLX�I<�<H@I]\ KLB �QXQM[B]> R H@I<OZX�J ! H_� P[ONx<H@K��%�0�<I<OZK[H_BEI H�PR�<H ��XQM[X�IqK[H�J:F6^_XJ�^@D�BEPLK*XQg�XQMNi 0.x<XQM[X21;BEI¢J�^@D�BEPLKCJ�^@^]KLMNH�J�I]\E^_X�P K[x]X����6I<OZK[H_BEI : H@P�J�^@D�BEPLK�^@H@I]X�J:M�> /K[H@^@H@I]\ 0.x<H@O¬x�^@H_X�PtOQ^_BEPLX�KLB : 0.H�^@^;^�H_XCOQ^@BEPLX�KLBSJR^@H@I]X�J:Mt�0�<I<OZK[H_BEI BEI7D7BEPLK KLMNH@J�I]\E^@X�PQ>R B%H_KCP[���,OZX�P�KLB�OZBE�<I;K�K[x]XI;�6DGF�XQMB��1K[H@^@H@I]\EP 0.x]BEPLX�x]X�H_\Ex;K�0�<I<OZK[H_BEI¢H@POQ^@BEPLX�KLBJ7^@H@I]X�J:Mz�0�<I<OZK[H_BEI�>

w.B60�XQg�XQMRJ7PNx]B�M[K�KLX�ONx6I<H@OQJ�^1^@X�D7D7J 3�X�P[PLX�I;K[H@J�^@^@inP[�<F6J��<�<H_K[H@g+H_K iB5*�<M[B�g�X�PYK[x<J:KH_� "²H�PYJ�KLMNH@J�I<\E�<^@J:M�M[XQ\EH_BEI 3�B�M.B�K[x]XQMI<H@OZXRM[XQ\EH@BEI 5W1<K[x]XGI+�<D%F�XQMYB��tK[H@^@H@I]\EP�0.H@K[xx]X�H_\Ex;Kn�0�<I<OZK[H@BEI�^_i+H�I]\/OQ^@BEPLX KLB|J��]ljX���^@H@I]X�J:M �0�<I<OZK[H_BEI��'BEI K[x<J:KkM[XQ\EH_BEI�H�PXZl+� 3+V�� X�I;K#3��#5��23 ? @ �B3 ?#5�5�5W1 0.x]XQM[X V H�P�K[x<XSJ:M[X�J B�� " J�I<� X�I;K63��#5H�P�J%�0�<I<OZK[H@BEI�]XQ��X�I<�<H@I]\ BEI<^_i¢BEIkK[x]XSPN^_B���XB����:>

aYx]X�XZlj�6^@H@OQH_KCOZBED��6�]K[J:K[H@BEI B��1X�I;K#3�# H%$ 5COZBED�X�P���M[BEDªOZBED��6�]K[H�I]\%K[x<X�I+�<D%F�XQMB��<K[H@^@H@I<\EP1BEI J�KLB�MN�<PH0.H_K[x J\EH_g�X�I P[^_B���XJ3�# H%$ 5W1�K[x<J:K H@P�1�J\EH_g�X�I x]B�MNH �QBEI;K[J�^jx]X�H_\Ex;KONx<J�I]\�Xed # � f1\�BEH�I]\7x]B�MNH �QBEI;K[J�^@^_inJ:M[BE�<I<�kK[x]XRKLB�MN�<P�1 J�I<�Gd

$) f�g�XQM[K[H@OQJ�^@^_i�>

a1B OZBED��6�<KLXK[x6H@P 0CXR�<PLX�P[K[J:\�\�XQM[X�� 0CX�H_\ExqK[P�� H 9 H * H ,BEI,K[x<XX��<\�X�P�B�� = > 1jJ�PH@IT�tH_\E�]M[XG�+>e � 0CX �6�]KYK[x]X�P[X 0CX�H_\Ex;K[P�H@I�K[x]X hSJ�PLKLX�^_XQijI D,J:KLMNHmlkK[x]X�I H_K[P��<XQKLXQMND7H@I<J�I;K�H�P

K[x]X%P[�6D B�g�XQM�K[H@^�H@I]\EP�B�� K[x]X 0�X�H@\ExqK�B��*J7K[H@^@H�I]\ 3��<M[Bj�<�<OZKYB�� K[x<XGX��]\�X 0�X�H@\ExqK[PW5¬>aYx<J:K H@P�1 1 2 �<XQK6� 2 H�P,K[x]X s�8<�����r����� � -f������������� 3�� H 9 H * H 5¢B��GK[H@^@H�I]\EP�0.H@K[x

Page 14: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�~:¥ ����� �����#���

a b

dc

c

c c

c

cd

d

dd

d

a

a

a

a

a

bb

b

bb

�tH_\E�]M[XG� ]�� �F� a2�, I¤=�2�� I�_�I¡�¤FaC�~�_£��

K[x]X�PLX"0CX�H_\Ex;K[PQ> �]B�M J�K[H@^@H@I<\nB��zK[x]X¢KLB�MN�<PE0.H_K[x K[x]X�P[X�0�X�H@\ExqK[P�1�K[x]XnI;�<D%F�XQMSB���]BED7H@I]BEP*B���K i;��X � H 9 H * H ��]XQKLXQMND7H@I<X�P K[x]X�x<X�H_\ExqK�ONx<J�I]\�XJ:M[BE�<I<�7K[x]XYKLB�MN�<P�]�K[x<Xx]B�MNH �QBEI;K[J�^�x<X�H_\ExqK ONx<J�I<\�X H�P 3�� - §�� , 5�, ) J�I<�2K[x]X¢g�XQMNK[H@OQJ�^�x<X�H_\ExqK ONx<J�I<\�X H�P3�� ) §�� + 5�, � >CaYx]XQM[XQ��B�M[XRBEI<OZX*0CX*:jI]B#0

3�� H 9 H * H 5 = [��������� ����

����� 9 ��� * ��� ����0�XGx6J�g�X

3�� H ?�, � H * H ?�, * 5 = [���������� ����

� I��� * � h ���

J�I<�G0CX¢I]XQX��2BEI<^_ipXZl+KLM¬J�OZK%K[x]XnJ:�<�<M[B��6MNH@J:KLX7OZB+X �,OQH_X�I;KGKLBp\�XQKGK[x]XnI;�<D%F�XQMSB��K[H@^@H@I]\EPCB��t\EH_g�X�IkPN^_B���X�>

efI d c^f1K[x]XGJ�PLijD%KLB�K[H@OR��B�MND%�6^@J�H@Pz\EH_g�X�I

^_B�\ 3�� H 9 H * H 5 =?

3 = > ; 5 > � _ � � _!� ^_B�\g3�� @ 9 ��@ * � @ � � 5 �� �� H�rM[BED 0.x<H@O¬x K[x]XGJ:F�B�g�XS��B�MND �<^@J ��B�MYX�I;KY�rBE^�^_B#0.P�>

Page 15: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�~j?

���� �"2 ���- �]B�M,K[x]X / �QKLX�O �<H@J�D�BEI<�F1*K[x]Xk�0�<I<OZK[H_BEI : ����� H�P7I]B�K,J�I<J�^@i;K[H@O^1�BEI<^_i/�6H_X�OZX�0.H�PLXJ�I<J�^_i+K[H@O:> efInK[x]X���BE�]MzM[XQ\EH_BEI<PCBE�]K[PNH@�]XB��1K[x]XRH@I<PNOZMNH_F�X��,OQH_M¬OQ^_X : ����� H@Pz^@H@I<X�J:MJ30J�Pi�BE�pOQJ�I'J:�<�6M[B�l]H@D7J:KLX�^_ikP[XQX H�IkK[x]X XZljJ�D7�6^_X K[H@^@H@I]\ 5¬>.efI<PNH@�]XGH_KH@PJ�I<J�^_i+K[H@O:>R�RIK[x]X¢F�BE�<I6�<J:M[i B��YK[x]XTH@I6P[OZMNH_F�X��9OQH_M¬OQ^_X 3�K[x]X 8<��������� �q�����q3�& 1CP[H@I<OZXnH_K PLXQ�6J:MNJ:KLX�PK[x]X$� ��M[B��QX�IpM[XQ\EH_BEI ,��M[BED K[x]X$� KLX�D7��XQM¬J:KLX� �QBEI]X65.K[x]X%�0�<I<OZK[H@BEI�H@P � �F6�]K�I<B�K� > >*e I ��J�OZK.H@KYH@P � ��� � XZl]OZXQ�<K.J:KYK[x<XS�rBE�<MYF�BE�<I<�<J:M[i 8 X��]\�XRD7H@�<��BEH�IqK[PQ>

aYx]X�PLX��rMNB��QX�I,M[XQ\EH@BEI<P*M[X�PN�<^_K*��M[BED K[x<X�]XQ\�X�I<XQMNJ:K[H_BEI¢B���K[x]X.X�^@^�H_�<K[H@OQH_K i%B���K[x<X{� � �]XQ� I<H@I]\ : ����� J:KYK[x]XGF�BE�6I<�<J:M[i7B��tK[x]XS�<BED7J�H@ITB��t�]XQ�6I<H@K[H_BEI�>

e K7H�P%g�XQMNi/H@I;KLXQM[X�PLK[H�I]\ KLB9P[K[�<�]iZ0�x<J:K�x<J:�<��X�I<P XZljJ�OZK[^@i/J:K�K[x]X F�BE�<I<�6J:M[i�>E XE0CBEI��dK.�<H@PNOQ�<P[PCK[x<H@P�x]XQMNXSK[x]BE�]\Ex�>

� ����-R��S'FOUY�V .¬���?Y?'F.�YY�-, �

aYx]X�PNOQJ�^@H@I]\ ^@H@D7H_KS30^�H@D7H_K 0�x]X�I7K[x]X�^@J:KLK[H@OZX�PL�6J�OQH�I]\GKLX�I<�6PKLB �QXQM[B 5B���K[x]X�x]X�H_\Ex;K���<I6OZK[H_BEI BEI��]BED,H@I]B K[H@^@H@I<\EPzKLX�I<�<P�KLBnJ7I<H@OZXSOZBEI]��B�MND7J�^�^_i¢H@I;g:J:MNH@J�I;KYOZBEIqK[H@I+�]BE�<P�<M[BjOZX�P[P�1EK[x]X �LD,J�P[P[^_X�P[PC��J��<P[P[H�J�I �rM[XQX.�<X�^@�� 1EJSP[B�M[K B�� K�0�B^8��6H@D�X�I<P[H@BEI<J�^jg�XQMNP[H_BEIB���CM[B#0.I6H@J�I D7B�K[H_BEI�>efI K[x<H@PPLX�OZK[H_BEI 0CX �6H@P[OQ�<P[P.K[x]X H@�]X�J�P�F�X�x6H@I<�TK[x]X �<M[B+B��f>aYx]XRB�MNH_\EH�I<J�^�M[XQ�rXQMNX�I<OZX�P.J:M[Xedm?�?211?�=6f >

%��� � � �� � �-� �S / �-��� ��E Xzx6J��%J^_B�K1B�� ONx]BEH@OZXzH@I �]XQ�6I<H@K[H_BEIRB��<K[x]XChSJ�PLKLX�^_XQijI D7J:KLMNHml�>1ef�]i�BE� \�B�F6J�O�:SKLBK[x]X��<M[B+B���1ji�BE�40.H@^�^6PLXQXRK[x<J:K?0CXROZBE�<^@� x<J�g�XSONx]BEPLX�I�J�Iqi¢X��]\�X�0�X�H@\ExqK[P!0.H@K[x¢K[x<X�<M[B���XQM[K�i K[x<J:K�K[x]XE0CX�H_\Ex;K[P.BEI J¢^�J:KLK[H@OZX%P�A+�<J:M[X�� H H � H �7P[J:K[H�PL�ri K[x]X OZBEI<�<H_K[H@BEIK[x<J:K�� � , �GH@PM[X�J�^�J�I<�¢I<XQ\EJ:K[H_g�X�>*efKCH@PCOZBEI;g�X�I<H_X�I;K�KLB �6�]K70�X�H_\Ex;K[PzJ�PzH@I,�tH_\E�]MNXU;> ! X�D7D,Jk?Rx]BE^@�<Pz��B�MYK[x]X�P[X*0�X�H@\ExqK[P�J�P�0CX�^@^�1]J�P.�]B7a.x]XQB�M[X�D7PS?GJ�I<�k=+>! XQK �<P'I]B60 P[K[�<�]i � ��� ��B�M K[x]X�PLX 0CX�H_\Ex;K[PQ> �CX�OQJ��<PLX B��7F6H@�6J:M[K[H_KLX�I]X�PNP�1� ��� 3 � � H � > 5 = ¥ �6I<^_X�P[P BEI]XTB�� � � H � > H@P F ^@J�O�:�J�I<��K[x]XTB�K[x]XQM¢[email protected]<H_KLX�> ! XQK� �]X�I]B�KLX'K[x]XpF6^@J�O�: g�XQM[K[H@OZX�PTJ�I<��� K[x]X 0.x<H_KLX g�XQM[K[H@OZX�PQ> ! XQK � 8�� > ef�: 8�����1]K[x]X�Ik� :�� ¥�D7X�J�I<P

: 3 � @ ?#5t§ : 3 � §|?#5 @ ; 3 : 3 � @ ; 5t§ : 3 � § ; 5�5 = ¥

K[x<J:K.H@P+� � � @ ; � � : = ¥ f E X%P[J�i : 8���� H@P �t�r5��E��&;��&p86��863r°������ H_��� : = ¥j>! XQK � � H � � F�XnK[x]X F6^@J�O�:2g�XQM[K[H@OZX�P�0.x]BEPLXTOZB+B�MN�<H@I<J:KLX�P�J:MNXnF�B�K[x�XQg�X�I 3�M[X@8PL��X�OZK[H_g�X�^_i21.F�B�K[x�Bj�<� 5¬> R H@D,H@^@J:MN^_i�^_XQK�� � H � � F�X K[x<X 0�x<H_KLXpg�XQM[K[H@OZX�P40.x]BEP[XOZB;B�MN�6H@I<J:KLX�PYJ:M[Xe3 ? H ¥�5 D�Bj�,= 1<M[X�P[��X�OZK[H@g�X�^_i 30¥ H ?#5D�Bj�7=+>

Page 16: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�~�= ����� �����#���

1 -1

i-i

-i

-i

-i

-i

-i

i

i

i

i

i

1

1

1

1

1

-1

-1

-1

-1

-1

�tH_\E�]M[X%U ]

� &j(u(u89!�� y : 8���� JK] MEJK]Vo,D�L,G~L ?Ea�?Eu{kRGXJXo,^ G*S;LVa : JK] S0?ED2UWBEa�JZo�BRa � ��?EaM � �],L�89?EDF?HG~LVu{k�^�G*S0?RGsJK]C^c : 3 9 5 = : 3 9 @ =�5 @ : 3 9 §u=�5 @ : 3 9 @ = ; 5 @ : 3 9 §u= ; 5 f

OQBED�L�BEbRLVD_^sG*S;LV],L�G 3 B�S0?ED2UWBEa�JZo�y2[;a�otGXJZBEa9]�?ED�L�oCBRa 4_[Hq ?RG~L @aYx]X�<M[B+B����rBE^�^_B#0.PC��M[BED K[x]XB�F6PLXQM[g:J:K[H_BEInK[x<J:K�� < ��H@P�K[x]X ! J:�6^@J�OQH�J�I"0.x<X�I

M[X�PLKLMNH@OZKLX��kKLB,J�Iqi B���K[x]XGP[�<F6^@J:KLK[H@OZX�P � � H � � H � � H � � 16PLXQX%��H@\E�]M[XS~+>/ P.J7OZBEI<PLX�A+�]X�I<OZXG� ��� OQJ�ITF�XRM[X�^�J:KLX��TKLB7K[x]X ��M[XQX�I���PY�0�<I<OZK[H@BEI�>

%�� � �±��� � ���I� � /��S��/�/ ��� ��L�$� �-� ���]B�M�J �]ljX�� � 1 �2���#3 � H 9 5*30OZBEI<P[H@�]XQMNX��TJ�PYJ��0�<I<OZK[H_BEInB�� 9 5�H@Pz�<H@P[OZM[XQKLXSJ�I6J�^_i;K[H�O^1XZljOZXQ�<K�J:K � >CaYx<H@PC��BE^@^_B#0�P��rMNBED �'�2��� = � >

efI'�0J�OZKS� ��� OQJ�I F�X 0.MNH_KLKLX�I'H@I'KLXQMND7P�B��CK[x]X,�M[XQX�I���PR���6I<OZK[H_BEI % > WYX�OQJ�^@^K[x<J:K$%D3 � H � 5tH@Pt�]XQ�6I]X�� KLBRF�XCK[x]Xz���<I6OZK[H_BEI P[J:K[H@PL��i+H�I]\© %�3 � H � 5 = ��� 3 � 5!3�K[J^:+H@I<\K[x]X ! J:� ^@J�OQH@J�I40YM[K�K[x<XSPLX�OZBEI<�kg:J:MNH@J:F ^_X65¬>*�]B�M�J7�]l+X�� � 1

©�� ��� 3 � H � 5 = � < �p� ��� 3 � H � 5 = 30� < � � 5@3�� 5 = � � ( � §� � ��� § ; 3�� � ( > § � � � > 5 f

Page 17: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�~�`

1−1

i

−i

1−1

i

−i

−1

−1

−1

−1

−i+i=00

0 0

41 −1

−i

i

i

i

i

−i

−i

−i

−1

−1

−11

1

1

K K*1

�tH_\E�]M[XG~ ]

aYx;�6P�� ��� 3 � H � 5�H@PJ¢PN�<D B���rBE�]MS�M[XQX�I���P��0�<I<OZK[H@BEI<PQ> E x]X�I M[X�PLKLMNH@OZKLX��pKLB � � 1H_K*H@P K[x]X��<H ��XQM[X�I<OZX�B���K 0CB �MNXQX�I<P*�0�<I<OZK[H_BEI6P�1q�u���#3 � H 9 5 = %�3 � @ ? H 9 5]§ %D3 � §? H 9 5 f%���� � � � ��/ � ��� � 032 /E X,OQJ�I9X�J�P[H�^_i OZBED�� �]KLX7�u������B�MGJ M[X�OZK[J�I<\E^_X21�PNH@I<OZX-0�X7J�^@M[X�J��]i'�<H@J:\�BEI<J�^@H �QX��K[x]XGD7J:KLMNH_ln�/>�CX�OQJ��<PLXG� [email protected]+D,D�XQKLMNH@O^1B0CXGx<J�g�XSK[x]XS��BE^@^_B60.H@I]\B]

� ��� 3 � � H � > 5 = [L: L 3 � � 5 : L 3 � > 5�,�� L

0.x]XQM[X : L J:MNXSK[x]XRB�M[K[x]BEI<B�MND7J�^@H �QX��kX�H_\�X�Iqg�X�OZKLB�M¬PS0.H@K[x X�H_\�X�Iqg:J�^@�]X�� L >efI/K[x]X ^@H@D,H_KE0.x<X�I2K[x]X M[X�OZK[J�I<\E^_X \�XQK[P�^@J:M[\�X21J�I<� � � H � > J:MNXn�0J:M �rMNBED�K[x<X

F�BE�6I<�<J:MNH_X�P�1 0CXGOQJ�Ik� I<�

� ��� 3 � � H � > 5 =?c > >

� >��� � >��

�� > d ��� ( �� e

= ; P[H�I�S§2=tP[H@I � � �� H 3 ?#5

0.x]XQM[X"3<; H = 5 = � > § � � 8 =?>Z>aYx<H@Pz��B�MND �<^@J �]XQ�6I<X�P�1<J�^_BEI]\D0.H@K[xTaYx]XQB�M[X�Db= 16J�D�X�J�P[�]MNXSBEITK[H@^@H�I]\EPCB���K[x<X�6^@J�I]X�>taYx<H@PtD�X�J�PN�]M[X�H@PtK[x]XY�<I6H�A;�<X70CX�J^: ^�H@D7H_K�B���D�X�J�PN�]M[X�PtBEI�M[X�OZK[J�I<\E^_X�PQ>te K*H�PJ�^@PLB K[x]X�6I<H�A+�]XYX�I;KLM[B��;i 8�D7J�l]H@D7H ��H�I]\SKLMNJ�I<PN^@J:K[H_BEI 8�H@I;g:J:MNH@J�I;K�D�X�J�P[�<M[X.BEInK[H@^@H�I]\EPdd=bf > R BED�XRg:J�^@�]X�PYB���� ��� J:M[XSP[x<B#0.IkH�IT��H@\E�]M[XG¤+>

/ P[i+D��6KLB�K[H@OQJ�^@^_i21<H_� � 8 � � 1 0�X%x<J�g�X

� ��� 3 � H 9 5 =�� WYX �� d + � � e @ � 3 �� + � � � � 5 9 8 � �; e D �� d + � � e @ � 3 �� + � � � � 5 9 8 � � f

Page 18: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�~�c ����� �����#���

-

-

-

-

-

- -

-

+-

-

-i4

0 − 13 10π

4 π )-i(

0

00

0 0

0

0

0

-i(

1

1 1

1

π

π

0

0

0

0

0

0

5π 4

-i(

-i( 13 10)

4

34

2

3)

(0,0)

4

-i(1 1 )

4

)

4

5 4

14 4π

4

14

4 π

π

�tH_\E�]M[XG¤ ]

aYx]X�PLXSJ:M[XGM[XQg�XQMNPLX��kH@� � 8�� � >

%���% � $� � / � �$ �� � ���$ E X%OQJ�IkP[H�D7H@^@J:MN^@i7OZBED��6�]KLXSK[x<XSJ�PLi+D7�<KLB�K[H@OQPYB��t�u���YBEITF�BE�<I<�]X�� �]BED,J�H@I<PQ>! XQK� F�X�J��EB�MN�<J�I �]BED7J�H�I�> ! XQK� > �F�X�J �]BED7J�H@I H@I = ��=?>�0.x6H@ONxnJ:�<�<M[B�ljH�8D7J:KLX�P %>te KzH@P�JE0CX�^@^98;:jI]B#0�I7�0J�OZK�K[x<J:K�K[x]X��<H@P[OZM[XQKLX��H_MNH@O¬x<^_XQKC�MNXQX�I���P�0�<I<OZK[H@BEIBEI > �?30�]XQ�6I<X�� F;ig] %�3<; H = 5 H@P K[x<X�XZl+��X�OZKLX��,I+�<D%F�XQM B���� J�P[P[J:\�X�PJ:K = B���JSP[H�D-8�6^_X7MNJ�I6�]BEDU0zJ�^�:9PLK[J:MNKLX��uJ:K ;|J�I<�2P[KLB��<��X��uJ:K%K[x]X¢F�BE�<I6�<J:M[i21tJ�^_KLXQMNI<J:K[H_g�X�^@i21%D3<; H = 5�H@PK[x<X�H@I;g�XQMNPLXB���K[x]X��<H@P[OZMNXQKLX ! J:� ^@J�OQH@J�I 0.H_K[x,��H_M¬H@ONx<^@XQKF�BE�<I<�<J:MNi OZBEI 8�<H_K[H_BEI<P5J:�6�<M[B�l]H@D7J:KLX�PYK[x]XGOZBEI;K[H@I;�<BE�<PY��H_MNH@O¬x<^_XQK.�M[XQX�I���PY�0�<I<OZK[H_BEI BEI >

E X,OQJ�I9�<P[X K[x6H@P���J�OZKGKLBkOZBEI<PLKLMN�<OZKSJ \�M¬J:�6x � 8 ��=?>��B�M*0.x<H�ONxpK[x]X7� ���B���XQM¬J:KLB�M�OZBEI;g�XQM[\�X�P�I6H@OZX�^_i,KLB,K[x]XGOZB�M[M[X�P[��BEI6�<H@I]\ OZBEIqK[H�I;�]BE�6P�B�F &LX�OZKYBEI >

Page 19: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�~�y

aYx]X \�MNJ:�6x �.x<J�P�J¢F ^@J�O�:kg�XQMNKLXZlp��B�M�X�J�ONx g�XQM[KLXZl'J�I<�p��J�OZX�B�� > �W1�J�I<� J0.x<H_KLXg�XQM[KLXZl �rB�M�X�J�O¬xTX��]\�X�B��� > � 30J�I<�� � H@PCK[x]XRP[�]��XQM[��BEP[H@K[H_BEI�B�� > � J�I<� H_K[P�<�<J�^�5¬>*w�B#0CXQg�XQM 0CXSM[X�D7B�g�XG��M[BED �BEI<XRF6^@J�O�: g�XQM[KLXZl 9 � BEITK[x]XGBE�]KLXQMY�0J�OZX�>

� &j(u(u8 o � S0?�] G*S;L ],?EU�L�a�[;U�@CL,D�BFy�@,u ?Ho ;�?Ea�M 3 SAJXG~L�bEL,D2GXJZo_LV] @aYx<H@P���BE^@^_B60.PSP[H�D��6^_ik��M[BED K[x<X � �<^@XQMR�rB�MND �<^@J�� §�� @�� = ?7J:�<�6^@H@X�� KLB > �¬>YefI �0J�OZKPL� J�I<I<H@I]\�KLMNXQX�P�B�� > �zM[B+B�KLX�� J:K 9 � J:M[X H@I�F6H &LX�OZK[H_BEI 0.H@K[xk��XQM[��X�OZK

D7J:K[ONx<H�I]\EP�B�� �J30PLXQX dm?�U 1�?Zc2f�5¬>! XQKS%D3 � H � 5TF�X9K[x]XuOZBEIqK[H@I+�]BE�<P ��H_MNH@O¬x<^_XQK �M[XQX�I���P��0�<I<OZK[H_BEI BEI %> ! XQK�%D3 � H � 5F�X K[x<X J�I6J�^_i;K[H�O%�0�<I<OZK[H@BEI B�� � 0.x<BEPLX M[X�J�^t�6J:M[KRH@P %D3 � H � 5¬>,��I �S0�Xx<J�g�X7� ��� = = � �% @ �B3��W5W11H@I K[x]X ��BE^@^_B#0�H@I]\ PLX�I<PLX�>G�XQ�6I<X �H� H � � J�I6� � ( H � �F;i

�%"3 � H � 5 = ?= 3 �H� 3 � H � 5! � � @ ;�� � 3 � H � 5! � � 5= ?c 3 � ( 3 � H � 5! � @�� � 3 � H � 5! �I5 f#%$�&;� ��&j( o�� S;L,aS2 � § 9 2�Je](a�BHG � 3��W5 3 L S0?EbEL

� ��� 3 � H 9 5 = ��� ������ L�� � 3 � H 9 5 @ �B3��W5 � 8 � ��?Ea�M 9 8� �� ; � U�� � 3 � H 9 5 @ � 3��5 � 8 � � ?Ea�M 9 8� ���� L�� � 3 � H 9 5 @ �B3��W5 � 8 � � ?Ea�M 9 8� �� ; � U�� � 3 � H 9 5 @ � 3��5 � 8 � � ?Ea�M 9 8� � f

�]B�M XZl]J�D��6^_XCBEI � > 0CXCx6J�g�X �%D3 � H � 5 = § �>�� ^@B�\ 3 � § �(5W1EPLB �% = �>�� - � � ( > - � �� � �J�I<� �H� =�� � = �� �� � � > / PzJ�I]B�K[x]XQMzXZljJ�D7�6^_X21jBEInK[x]X�<�<��XQM�x<J�^_��� ^@J�I]XS0CXRx<J�g�X�%D3 � H � 5 = § �>�� ^_B�\ � � �� ���� H ��M[BED 0.x6H@ONx40CXR\�XQK

�%�3 � H � 5 =?= >

T �� § � § �

� §�� Y f/ PYJ,OZBEI<PLX�A+�]X�I<OZX � ( 3 � H � 5 = >� d � � � e J�I<� � � 3 � H � 5 = § >� d � ���� e fe K�H@P X�J�P[H_XQM�KLBE0CB�M�:D0.H_K[x K[x<X.�0�<I<OZK[H_BEI<P ��� =��H����� � PNH@I<OZX�K[x]XQi�KLMNJ�I6PL�rB�M¬DI<H@OZX�^_i�>*W.X�OQJ�^@^�K[x<J:K �% H�P�OZBEI]�rB�M¬D7J�^@^_i,H�Iqg:J:MNH@J�I;K�1]K[x<J:KYH@P�H_� � ]�� � ¨H@P�J�OZBEI 8�rB�MND,J�^�x]BED7XQBED�B�M[�6x<H�P[D K[x]X�I �%! 3 � H � 5 = �% � 3 �H3 �\5 H � 3 � 5�5 f / PJnOZBEI<PLX�A+�]X�I<OZX� � J:M[X%OZBEI]�rB�M¬D7J�^@^_inOZB�g:J:MNH@J�I;K�]

� ( 3 � H � 5 =���( 3 �H3 �(5 H �H3 � 5�5 �#" 3 � 5 f� � 3 � H � 5 =���� 3 �H3 �(5 H �H3 � 5�5 � " 3 � 5 f

Page 20: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�~�� ����� �����#���aYx<H@P�J�^�^_B#0.P��6P�KLB,OZBED��6�<KLXGJ�PLijD��<KLB�K[H@OQP�B�� �u����BEI�J�I;i �EB�MN�<J�Ik�<BED7J�H@I�>­kB�MNX��<M[X�OQH@PLX�^@i21�K[x<X�M[X�P[�<^_K�H@PK[x]X���BE^@^_B60.H@I]\]> E x]X�I � H 9 J:M[X y,?ED�? 89?ED2G 1 0�XOQJ�I �<PLX K[x]X J:F�B�g�X7J�PLi+D7�<KLB�K[H@O �rB�M¬D%�<^�J�PQ> E x]X�IpK[x]XQi J:M[X�OQ^_BEPLX KLB�\�XQK[x]XQM�1�K[x<X

�]BED7H@I<J�I;K�KLXQMNDbH@PYOZBEI;KLM[BE^@^_X���F;iG3 ?#5?0.H_K[xkJ¢�]XQg+H�J:K[H_BEITB�� B�MN�]XQM��Y\EH_g�X�I F+inK[x<X���<I6OZK[H_BEII30PW5 � ( 3 � H � 5*§ >� d � � � e J�I<� � � 3 � H � 5¬>%���� �/ � 0 � � � �� / � �w.XQM[X H@PJ,P[J�D7�6^_X%OZBED7�6�]K[J:K[H_BEI\1�K[x]X XZl+��X�OZKLX��pg:J�^@�]XGB�� �I3 ' 5 � 3��25�H@I�K[x]X �]�<��XQMx<J�^_�1� ^@J�I]X�>! XQK � > H 9 > F�XK[x]XRx]B�MNH �QBEI;K[J�^�X��]\�X�PzBEI J g�XQM[K[H@OQJ�^��6J:K[xn��M[BED ' KLB K[x]X ; 8�J�ljH@P�1K[x]X � > F�X�H@I<\7K[x]BEPLX-0.H_K[xpJ 0.x<H_KLX ^_XQ�rKg�XQM[KLXZl(1�J�I<�'P[H@D7H�^@J:MN^_in^_XQK * > H > F�XGK[x]BEPLX�rB�M�� 1<PLXQX%��H@\E�]M[X7?Q¥j>

a

a

a

b

b

b

c

c

d

d

p

q

2

11

1

2

3

3

2

2

1

�tH_\E�]MNX7?Q¥ ]

E X%x<J�g�X �I3 ' 5 = c�� > � > § 9 > 1 �I3���5 = c�� L * L § L > R B� 3��H3 ' 5 �H3���5�5 = ?�� [

>�L� 3�� > * L 5t§ � 3 9 > * L 5�§ � 3�� > L 5 @�� 3 9 > L 5 f

� J�ONx,B���K[x]X�PLX�KLXQMND7P�OQJ�I�F�X?0YMNH_KLKLX�I,J�P K[x]X.�]XQKLXQM¬D7H@I<J�I;K B���J%=!CG=RD7J:KLMNH_l 0.H@K[x

Page 21: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�~EU

X�IqKLMNH@X�PYH@IT�u���Q>�]B�M.XZl]J�D��6^@X

� 3�� >G* L 5 = �]XQK�� � ��� 3�� d � e> H � d ��e> 5 � ��� 3 * d � eL H � d ��e> 5�u����3�� d � e> H * d ��eL 5}�u���#3 * d � eL H * d ��eL 5��

0.x]XQM[X � d � e> H � d ��e> J:M[XCK[x]X�0.x<H_KLXCJ�I<�%F6^@J�O�:Gg�XQM[K[H@OZX�P�B�� � > 1EJ�I<� ^@H�:�X�0�H@PLX*��B�M * L >�aYx<X�<H@J:\�BEI<J�^jKLXQMND,P 0.H@^�^;J�^@^]OQJ�I6OZX�^<BE�]K�1+J�I<�7J:��KLXQMJSOZXQM[K[J�H@I,J�D�BE�<I;K*B�� MNX�J:M[MNJ�I]\EH@I<\B1BEI]XS�6I<�6PCH�I K[x]XG^@H@D,H_K

� 3�3�� 3 ' 5�§ �I3 ' 5�5@3��I3���5 § � 3���5�5�5 = ���� �

�������

¥ � ( 3 � � H � > 5� ( 3 � > H � � 5 ¥ ���� � � � >§ ���� �

�� ����

¥ � � 3 � � H � > 5� � 3 � > H � � 5 ¥ ���� � � � >§0���� �

������

¥ � � 3 � � H � > 5� � 3 � > H � � 5 ¥ ���� � � � >@ ���� �

������

¥ � ( 3 � � H � > 5� ( 3 � > H � � 5 ¥ ���� � � � > f�]B�M.K[x<XS�]�<��XQM�x<J�^_��8 �6^@J�I]XJ0CXSx<J�g�X � ( 3 � � H � > 5 = >� d � � � � � e J�I<� � � 3 � � H � > 5 =§ >� d � � � � � e >{�^@�]\�\EH@I<\ K[x<X�PLXGH@I \EH_g�X�P

§ c

> >��������

?3 � > § � � 5 >

� � � > @ c

> >��������

?3 � > § � � 5 >

� � � > @@ c

> >� ��� ��

?3 � > § � � 5 >

� � � > § c

> >� ��� ��

?3 � > § � � 5 >

� � � > faYx]XR�<M¬PLK�B���K[x<X�PLXGH@I;KLXQ\�MNJ�^@P�\EH@g�X�P

§ c

> >^_B�\ 3 ' § ��5@3�S§ #b5

3 ' §�#b5@3�R§ ��5 faYx]XQM[XQ��B�M[X

� 3�3��H3 ' 5z§ �H3 ' 5�5@3��I3���5*§ �I3���5�5�5 = c

> >T §.=:W.X�^_B�\ 3 ' § �25@3�R§�#^5

3 ' § #b5@3�R§ ��5 @

@ =:W.X1^_B�\ 3 ' § �25@3�S§ #b53 ' § #b5@3�S§ ��5 Y= ~

> >WYX�^_B�\ T ' § �

' § � Y f� B�KLXTK[x6J:K K[x6H@P A+�<J�I;K[H_K�iuH@P J D �<^_K[H_� ^_X7B��.K[x<Xn��H_MNH�ONx<^_XQK7�M[XQX�I���P%�0�<I<OZK[H@BEI%D3 ' H ��5¬>

Page 22: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�~�~ ����� �����#���%���� � �1/ �S� $ � � � �0��/�/�� / 2 �! XQK � � 3 ' 5 = �I3 ' 5 § � 3 ' 5zF�XSK[x<X �6�<OZK[�6J:K[H_BEIkB�� � J�0zJ�i �rM[BED K[x]X%D7X�J�I�>Y­kB�M[X\�X�I]XQMNJ�^@^_i'BEI]X,�6I<�<PRK[x6J:K%J�^@^ K[x<X,D�BED�X�I;K[PGB���K[x<X,x]X�H_\Ex;K �6�<OZK[�6J:K[H_BEI<PGOQJ�IuF�X0YMNH_KLKLX�ITH@I KLXQM¬D7PYB��*�M[XQX�I���P.���6I<OZK[H_BEI<P�]

#%$�&;� ��&j( � >}LtG @_L�?��;BRDFMH?RaNMHBEUW?EJ*a 3 JXG*S(]_UWBtBRG*S6@�BE[;a�MH?RD_k @ >}LtG ' � H fFfFf H ' � 8 @CL�MEJK]2GXJ*a�opG 8;BEJ*a\G*] @ � y c JK]�BtMHM 3 L�S0?EbRL ^@H@D�� ! � � 3 � � 3 ' � 5 � � � � � 3 ' � 5�5 = ¥Z@ � yc JK]�LVbEL,a 3 L S;?EbEL^�H@D� ! � � 3�� � 3 ' � 5 � � � � � 3 ' � 5�5 =UT § ?��

> Y ��� > [8;?EJ*D_J*a qR] ] %�3 ' ]�d ��e H ' ]-d > e 5 � � � %D3 ' ]�d � ����e H ' ]�d � e 5 f�Ci E H@O�: ��P1K[x]XQB�M[X�D 1�K[x<H@P�H@D7�6^@H_X�P�K[x<J:K � � 3<;(5�H�P�K[x]X*�<I<H�A;�]X��J��6P[P[H@J�I��<MNB+OZX�P[P

0.H_K[x OZB�g:J:MNH@J�I6OZX7���6I<OZK[H_BEI � 3�� � 3<;(5 � � 3 = 5�5 = %D3<; H = 5¬>,aYx6H@PROQJ�I F�X�K[J^:�X�I2J�PSJ�]XQ�6I<H_K[H@BEI¢B��tK[x]X (u8<5�5�3r&+5�5� 86��5�5���86�±-���&;&��&j3�� >

/ I�J�^_KLXQMNI<J:KLXG�<X�P[OZMNH_�<K[H@BEITH@P�J�PY��BE^@^_B60.PQ>��I��^@XQK � � > � F�X�J�I�B�MNK[x]BEI]B�MND7J�^jX�H@\�X�IqF6J�PNH@PtB�� K[x<X ! J:�6^@J�OQH�J�I 0�H_K[x���H_MNH@O¬x 8

^_XQK�F�BE�<I<�<J:M[inOZBEI<�<H_K[H@BEI<P�1]J�I<�k© � >�= � > � > >�XQ�6I]X

���t��3<;(5 = #[> M �

* >1 2 � > 2 � > 3<;(50.x]XQM[X * > J:M[XSH@I<�<XQ��X�I6�]X�IqK.��J��6P[P[H@J�I<PzB��tD�X�J�I ¥7J�I<� g:J:MNH@J�I<OZX,?:>

aYx<H@P PLXQM¬H_X�Pk�]XQ�6I<X�PTJ �<H@PLKLMNH@F6�]K[H_BEI\1YI<B�K�J����6I<OZK[H_BEI\]�K[x]X9PLXQM¬H_X�PT�<H_g�XQM[\�X�PJ�^@D�BEPLKTP[�]MNX�^_i|J�^@D�BEPLK XQg�XQM[i 0.x]XQMNX�> w�B#0CXQg�XQMk�rB�MkJ�P[D7B;B�K[x �0�<I<OZK[H_BEI � K[x<XPLXQMNH_X�P

���t��3��!5 -�� �= #[> M �

* >1 2 � > 2 � � � > 3<;\5��T3<;(5! ;OZBEIqg�XQM[\�X�PJ�^@D�BEP[K.P[�]M[X�^@i�>

#%$�&;� ��&j( v0/21r +!,365�� ] � G~LVa�ME] GIB ¥ ^ � � G~L,a�M�] 3 L�?";�u{k J*a MEJK]2GXD2JI@V[9GXJXBRa GIB c , > >GXJKU�L2]sG*S;LsUW?�]C]2u LV]C].y_D�LCL��sLVu M % ��� BRa JKaWG*S9L�],L,a9],LjG*S;?RGEyVBRDs?Ea�k ]2U6BpBRG*S�y2[;a�o,fGXJXBEa � BEa ^�G*S;L6Dn?Ea�MABEU b�?RD_JZ?A@Vu L ��> � � ^ � � �T3<;\5 � � 3<;\5 G~LVa�M�]�J*a MEJK]2GXD2JZ@,[;GXJZBEaGIB��� � % ��� 3��?5 fe K H@PE0�B�M[K[x/��BEH@I;K[H@I]\kBE�<K%K[x<J:K�1*J�^@K[x]BE�]\Exug�J�^@�<X�PGB���� � J:M[X H@I;KLXQ\�MNJ�^J30J:�rKLXQM

J��<�<H@I]\ J|I<BEI 8 MNJ�I<�]BED PND�B;B�K[x ���<I6OZK[H_BEI � 5W1 � � OZBEIqg�XQMN\�X�P�KLB¡J¡OZBEIqK[H@I+�]BE�<PB�F & X�OZKQ>

Page 23: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�~�¤

aYx]XQM[X%H@PYJ,^@B�K.D�B�M[X%K[x<J:K�OQJ�I�F�X%P[J�H@�kJ:F�BE�]KYK[x<X%�<H@P[KLMNH_F6�]K[H@BEI,B�� K[x]XGg�J�^@�<XB�� �kJ�I6� � � J:KYJ ��BEH@IqK�1jB�M�K[x]XL&LBEH@I;KC� �<I<OZK[�<J�^��<H@P[KLMNH_F6�]K[H@BEI<PQ>1efK�OQJ�ITF�X��<M[B�g�X��K[x<J:K7K[x]X��6H@PLKLMNH_F �]K[H_BEIuB�� �±J:K,J ��BEH@I;K�KLX�I<�<P�1<0.x<X�I|P[OQJ�^_X�� F+i�� ^_B�\ �� 1�KLB2J��J��<P[P[H�J�I40.H_K[xTg:J:MNH@J�I<OZXRX�A+�<J�^�KLB¢PLBED�XG�<I6H_g�XQMNP[J�^�OZBEI<PLK[J�I;KQ>

� ��� �n� '\, � V�YC�(.[�� ��� , 'F.L�(.�, Y�V �DV�YY�-Y?' XS'\Y � M �

� � KLB�I]B60 0CXzx<J�g�XCF�XQX�I*0CB�M�:+H�I]\YBEI P[�]F<\�MNJ:� x<P�B�� = > >t�]B�MtK[x]X��<H�D�XQM1D�Bj�]X�^EBEIOZXQM[K[J�H@I'B�K[x]XQMSMNXQ\E�<^@J:MR\�MNJ:�6x6P�1�P[�<O¬x J�PRK[x]X�x]BEI]XQijOZBEDGF2\�MNJ:�6x\1�P[H�D7H@^@J:MM[X�P[�<^@K[POQJ�I�F�XpB�F6K[J�H@I]X���> w�B#0CXQg�XQM ��B�MkD�B�M[X \�X�I]XQM¬J�^R��XQMNH_Bj�<H@Ok� ^@J�I<J:MT\�MNJ:�6x6P�1�K[x<XP[H_K[�<J:K[H_BEI¢OQJ�I,F�X.D7B�M[X�OZBED�� ^@H@OQJ:KLX���>te I7K[x<H@PPLX�OZK[H_BEI 0�X�<H@P[OQ�<PNP*JG�<H ��XQM[X�I;KF �]KM[X�^@J:KLX�� D�Bj�]X�^�1+K[x]XR��h�8 ��XQM¬OZBE^@J:K[H_BEI D�Bj�]X�^�1+BEInJ�ITH�IqKLXQM[X�P[K[H@I]\ ��J�D7H�^_i�B����6^@J�I6J:M\�MNJ:�6x<PtK[x]XC�LH@PLB�MNJ��<H�J�^ \�MNJ:�6x<PQ>t�]B�M�K[x<H@Pt�0J�D7H@^_iSB���\�MNJ:� x<P 0�X�x<J�g�X.JGP[�]M[�<M¬H@P[H@I]\�<M[B���XQM[K�ipB��CK[x]X��6J:M[K[H_K[H@BEI'���<I6OZK[H_BEI J�I<�9D�X�J�PN�]M[X�PRK[x<J:KGK[x]XQip�<XQ��X�I6�'BEI<^_i�BEIK[x]XG^_BjOQJ�^�PLKLMN�<OZK[�<M[XRB��tK[x]XR\�MNJ:�6x\1<I<B�K.H_K[PY^_BEI<\^8 MNJ�I]\�XRB�MN�]XQM�>

efIuK[x]XTP[�]F PLX�A;�<X�IqK PLX�OZK[H_BEI 0�X M[XQK[�<MNI2KLB �<H�D�XQMNPSF �]K%J:\EJ�H@I�BEI/K[x<H�PGP[J�D�X��J�D7H�^_i,B��tH@PLB�MNJ��6H@J�^�\�MNJ:�6x<P�>

efIuK[x<H@P%PLX�OZK[H_BEI 0�Xe0.H@^�^ P[x]B#0 K[x<J:K K[x]X,� J:M[K[H_K[H_BEI2���<I6OZK[H_BEI9B���K[x]XnMNJ�I<�<BEDOQ^@�<PLKLXQM�D�Bj�]X�^1x<J�PJ¢�6J:M[K[H@OQ�<^�J:MY�rB�M¬D ��B�M�K[[email protected]��X�OQH@J�^1��J�D,H@^_i B��*�6^@J�I6J:M.\�MNJ:�6x6P3�0.x<H@O¬xTH@I<OQ^@�<�<X�P�K[x<XGOQJ�PLXE= > 5¬>

WYX�OQJ�^@^�K[x]XS�]XQ�6I<H@K[H_BEI,B���K[x]XS�thJ8 ��XQMNOZBE^@J:K[H@BEIG3�MNJ�I<�]BED OQ^��<PLKLXQM5�D�Bj�]X�^ > ! XQK% F�X�Jn\�MNJ:�6x�>GaYx]X�PL�6J�OZX B��zOZBEI]�<\E�]M¬J:K[H_BEI<P�H@P�� =B��� ] � � � ¥ H ? ��� 3�X�J�ONxX��]\�XGH@PzB���X�ITB�MOQ^_BEPLX�� 5¬>! XQK � ] � � 3 ? H 5�F�XSJ�I�J�P[P[H@\EI<D�X�I;K�B��H0CX�H_\Ex;K[PYKLB¢K[x]XSX��]\�X�P�1�J�I<� � �|¥J7OZBEI<PLK[J�I;KQ>

E X%�]XQ� I]XRJ��<M[B�F6J:F H@^@H_K i,D�X�J�P[�]MNXRBEI��Z1

� 3 � 5 = ? � , K

�� ���� �F3��b5 H

0.x]XQM[X * H@P K[x]X9I+�<D%F�XQM B�� OZBEI<I]X�OZKLX�� OZBED7��BEI<XQK[PTB��%B���X�I X��]\�X�PkB�� � 1K[x<X�<M[Bj�<�<OZK�H�PB�g�XQMYB���X�I,X��]\�X�PzB�� � 1]J�I<� H@PCJ I]B�MND7J�^@H ��H@I]\GOZBEI<P[K[J�IqK�1<OQJ�^@^@X��¢K[x<X�6J:M[K[H_K[H_BEI �0�<I<OZK[H@BEI\]

= [� ^�� � , K�� ���� �F3��65 f

� B�KLX�K[x<J:K70�x]X�I � = ?�0CX.J:M[X�M[X��<�<OZX��7KLBGK[x]X.P[K[J�I<�<J:MN����XQM[BjOQ^@J:K[H_BEI,D�B+�<X�^ >­kB�M[X,\�X�I]XQMNJ�^@^_i 0.x]X�I � H@PRJ�I2H@IqKLXQ\�XQM%K[x]X,D�Bj�]X�^ H@P�X�A+�<H_g:J�^_X�I;KRKLBkK[x]X �68�PLK[J:KLX

Page 24: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�¤:¥ ����� �����#���{�B�KLK[PzD�Bj�]X�^Idm?f4�jH@I7J�InJ:�<�<MNB��<MNH@J:KLXY^@H�D7H_K � � ¥SK[x]X�D�B+�<X�^6H@P X�A+�<H_g:J�^_X�I;K*KLBGK[x<XPL�6J�I<I<H�I]\%KLMNXQXGD�Bj�]X�^ >

�]B�M.�0�]M[K[x<XQMYH@I]��B�MND7J:K[H_BEI BEITK[x<XGD�Bj�]X�^�PLXQX d�U6f >

��� � ���2�� ��+e ��% H@P*�6^�J�I<J:M�^_XQK % < F�X.H_K[P�6^@J�I<J:MC�<�<J�^ > ! XQK � < F�X�0CX�H_\Ex;K[PBEI¢X��]\�X�P � < B�� % < 1�]XQ�6I]X�� F+i � < 3�� < 5 = ��, �F3��b5¬>� &j(u(u8'� l}BED�?Ra�k oCBEa �!qR[;DF?HGXJXBEa BFy�LCMtq LV]�BFy % 3 J*G*SC* oCBRUX89BEaLVa�G*]C^ c L�Mtq0L2]?Ea�M otkRotu LV]C^ 3 L6S0?EbEL!* § @ c = � ^ 3 S;LVD�L � JK]�G*S;LWa�[;U�@CL,D6Bny�bRLVDVGXJXo2L2]WBFy% @aYx]X�I 0�X7x6J�g�X�K[x]X ��BE^@^_B#0�H@I]\ H@�]X�I;K[H_K�ikF�XQK�0�XQX�I J�I<� < 1�K[x]X��<�<J�^��6J:M[K[H_K[H@BEI���<I6OZK[H_BEI\]O ��� s*�65������r���¨

< = � ��� � , � H3 S;LVD�L � JK](G*S;L�8�DnBtME[�otGdBFy�?EuKu!BFy G*S;L6LCMtq L 3 L,J qRS G*] � =��� ^ � �F3��65 @O ����� -QP R �]�<��BEPLX V H@P�J OZBEI]�<\E�]MNJ:K[H@BEI B���B���X�I X��<\�X�P�1\0.H_K[x * OZBED���BEI]X�IqK[P

J�I<� 0CX�H_\ExqK � , �� ^�� �F3��b5 f aYx]Xk�6�<J�^YOZBEI]�<\E�<MNJ:K[H_BEI|x<J�P7X��]\�X�P �ª§ V J�I<� * "

OZBED���BEI]X�IqK[P�>*e K[P�0�X�H_\Ex;K.H@P

� ,�� K� ^ � � � � < 3�� < 5 = � ,�� K

� ^ � � � ��F3��b5 = � ,�� � � � �� K

� ^�� �F3��65 H0.x]XQM[X � H@P K[x]X �<MNB+�<�6OZKnB�� J�^@^�X��]\�XZ0�X�H_\Ex;K[PkB�� % > �z�]KTF+i±K[x<X ! X�D7D,J 1*H@ V|§ * " = ��1]PLB�K[x]XJ0CX�H_\ExqK�B���J��<�<J�^�OZBEI]�<\E�]M¬J:K[H_BEITH@PzJ OZBEI<PLK[J�I;K�� � � , �K[H@D�X�P�K[x]X�0CX�H_\Ex;KYB��tK[x]XR�<MNH�D7J�^�OZBEI]�<\E�]M¬J:K[H_BEI�> �

efI¢�6J:M[K[H�OQ�<^@J:M*K[x]X��<�<J�^�H_K�i�H@P�J�I ��5���(2� ��st$��r5�( F�XQK�0�XQX�InK[x]X�OZB�M[M[X�PL��BEI<�6H@I]\�<M[B�F6J:F6H�^@H_K�i PL�6J�OZX�P�> / PzJ OZBEI<PLX�A+�]X�I<OZXT0�X�x<J�g�X ' � @ ' � � = ?^1 0�x]XQM[X ' � H ' � � J:M[XK[x]XSX��]\�XS�6M[B�F6J:F6H@^�H_K[H_X�PQ>

�� � ��� ���-� �L �0 ���� ���-� �E XGOQJ�IkPLBED�XQK[H�D�X�PzKLMNJ�I<PL��B�MND % 0�x<H@^_X�<M[X�PLXQMNg+H@I<\%K[x]XSD7X�J�P[�]M[X � > R XQXR�tH_\E�]MNX?�?:>

��H_g�X�I JuKLMNH@J�I]\E^_X'H@I)% 0.H_K[x|g�XQMNK[H@OZX�P � � H � > H � J�I<��X��]\�X 0�X�H_\Ex;K[P � > =� H � � = 9 H � � > = * 1�J�I<� J� V¦¢�#0.H@K[xRK[x]X�PNJ�D�X�g�XQMNK[H@OZX�P�J�I6�SX��]\�X�0�X�H@\ExqK[P V H � H � 130J�P.H@IT�tH_\E�]MNX7?�?#5�0�X%x<J�g�X

Page 25: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�¤j?

ab

c

A

C

B1 2

3

vv

v

v

v

1

3

v2

�tH_\E�]MNX7?�?^]

���V���t  ��F� a2�, %� ��� �.�F� a2�, }������ ��� � � � � �F�2�����F�� I�F� � ��������� ��������� ������� ������������ !���"�����# !���" !���"�# � � � � � �F�_�����F�� I�n� � ��������� ��� ��$ � $ � �

%� �&�������'�� (���"�'�*),+ $,- + $ ) - +

%� � � � �F�_�����F�� I�n� � ��������� ��� �.$ � $ ���

%� �/�������'�! ����0 ��*) -1$1- + $ ) - +

%� � � � �F�_�����F�� I�n� � �����2��� ��� � $ �3$ � �

%� �/�������'�� !���4 ��*) -1$ ),+ $ ) - +

%���_���m�F�_�����F�� I�n� ��� �.$ � $ �%

� � � $ � �3$ � �3$ � � �%65

�/�������'�� �� ) -7$ ),+ $,- +%

� ) - +%65

w.XQMNX,K[x]X7XQg�X�I;K[P%BEI9K[x]X,^@XQ�rKSMNXQ�rXQM%KLB OZBEI<I]X�OZK[H@BEI<PSgjH@J B���X�I9X��]\�X�P BE[9G*]_JZMALK[x]XYKLMNH@J�I<\E^_X�>�a.x]XYOZX�IqKLM¬J�^6OZBE^@�<D,I \EH_g�X�P K[x]X.J��6�<H_K[H_BEI<J�^\30D �<^_K[H_� ^@H@OQJ:K[H_g�X65I0CX�H_\ExqK�<�]XYKLB K[x]X.��BEP[PNH_F6^_X�B���X�I7X��<\�X�PCH�I<P[H@�]X�K[x<X.KLMNH@J�I<\E^_X�> aYx]X�^@XQ�rKOZBE^��<D7I7\EH@g�X�P�K[x<XJ��<�<H_K[H_BEI6J�^ 30D%�<^@K[H_�6^@H@OQJ:K[H@g�X65<0�X�H_\Ex;K��6�]XSKLB,K[x<XG��BEP[P[H_F ^_X�B���X�I�X��]\�X�PH@I<P[H@�<XK[x<X¦,>

aYx]X�¦�§ © KLMNJ�I<P[�rB�MND,J:K[H_BEIn�<M[X�PLXQM[g�X�PzK[x<X�D7X�J�P[�]M[X�PCBEI OZBEI<�<H_K[H@BEI¢K[x6J:KCK[x<X0�X�H_\Ex;K[PzH@I¢K[x]X © �+OZBE^��<D7I¢F�X��<M[B���B�M[K[H_BEI6J�^<KLB K[x]XJ0�X�H@\ExqK[PzH@I¢K[x]X V¦¢�jOZBE^@�<D,I�>aYx<H@Pz\EH_g�X�PYc7��BE^_i+I<BED7H@J�^�X�A;�6J:K[H_BEI<P���B�MYK[x]X�0CX�H_\Ex;K[P � H 9 H * H V H � H � >� &j(u(u89v |}S;LV],L6LCY,[�?RGXJZBEa9](S0?RbEL�?�]VBEu{[9GXJXBRa4BEa�u{k�J y

§ � @ � 9 @ � *1@ 9 * @ � 9 *X= ¥ 3 ?#5

?Ea�M§�� > § �B3+V @ � @ � 5 @ V � �?= ¥ f 3 =�5

� a G*S Je]�o_?�],L�G*S;L ],BEu{[;GXJZBEarJK]V = �� H � = �

9 �?= �* f 3 `�5

Page 26: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�¤�= ����� �����#����]B�M OZXQMNK[J�H@I K�i+��X�PTB��%\�M¬J:�6x<P��60CX�H_\Ex;K[P�1�K[x]X�PLX KLMNJ�I<P[�rB�MND,J:K[H_BEI<P K[J^:�X2BEI�J

�6J:M[K[H@OQ�<^�J:MN^_i,I<H�OZXR�rB�MND ]

���� & ' ��-� �$����2 /g��� / ���L���10� aYx<H@P*I<B�K[H_BEI,H@P��<�]X�KLB%�����,I ddybf�J�I<� ­�XQMNOQJ:KJdm?��^f > E XP[J�i % x<J�P�J�I ��5��6��8<�t��863&+(')*&+�����0��4 H_��%ªH�PR�]MNJ�0�IpP[BTK[x<J:KRX�J�ONx2��J�OZX7H@P�JTOZijOQ^@H@O ��BE^_i+\�BEI\1�K[x<J:KSH@P�1�H�PH@I<P[OZMNH@F6J:F6^_XH@ITJ7OQH_MNOQ^@X21<J�I<�kJ�^@^�K[x]X%OQH_MNOQ^_X�P�x<J�g�XGMNJ��6H@�<P�?:>

+ +

+

��H@\E�]M[X7?�= ]�� ¤Z�_�~�_��� �W_ �F� �R�F�������6a��C¥\��a_�~�C£��

efInK[x<H�PCOQJ�P[X�0�XROQJ�I X�D%F�X�� % < H@P[B�MNJ��<H@J�^@^@i J�P!0CX�^@^�1j�<P[H�I]\GK[x]XROQH_M¬OQ^_XOZX�I;KLXQMNP30J:Kz^@X�J�PLK�J�PC^@BEI]\ J�PzK[x<XOQH_M¬OQ^_XOZX�I;KLXQMNP�J:M[XRH@I,K[x]XRH@I;KLXQMNH_B�MCB���K[x]XROZB�M[M[X�PL��BEI<�6H@I]\��J�OZX�P B�� %E5¬> � J�ONx X��]\�X'x<J�P J2M¬x]BEDGF �<P,J:M[BE�6I<�±H@K"0.x]BEP[X g�XQM[K[H�OZX�PTJ:M[X K[x<Xg�XQM[K[H@OZX�PYB��tK[x]XRX��<\�XGJ�I<�kH_K[P.�<�<J�^ 30�tH_\E�]M[X7?�`�5¬>! XQK F�XRK[x]X%x<J�^_��8�J�I]\E^_X�B��tK[x]XSMNx]BED%F6�<P�J�I6�T�]XQ�6I]X

�F3��b5 = > � PNH@IF3 >��� 5P[H@I\3 >��� 3 � > § 5�5

0.x]XQM[X = OZBEP ��� 3 > ��,:=�5 faYx]X�I BEI]X%D7J�iTONx<X�O�:TK[x<J:K

Page 27: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�¤�`

1 1

1 1

eθ1

v2

v

�tH_\E�]MNX7?�` ]

� &j(u(u8 ¯ � ¦n§.© GXDn?Ea9]Ky,BED2UW?RGXJXBRa#8�D�L2]tLVD2bELV] Je],BEDn?HMEJZ?Eu JXGXk [;a�MAL,D�G*S;LV],L 3 L,J qRS G*] @aYx<H@PG�rBE^�^_B#0.P%��M[BED��6^@�]\�\EH@I<\�H@I9K[x<X 0CX�H_\Ex;K[P �F3��65GH@I;KLB 3 ?#5W1J�I6�u�<PNH@I]\�K[x<X

��J�OZKYK[x6J:K.J�KLMNH@J�I<\E^_XRXZljH�PLK[P�BEI<^_i 0.x]X�I ) @ + @ , = > ,:=+>���% � / ��� �$���50��-� �L / �6^@J�I<J:M�\�MNJ:� x % H@P s*&+�:�r���t��� H_� KLMNJ�I<PN^@J:K[H_BEI<PH@I = > J:MNX 3�0CX�H_\Ex;K 8 �<M[X�PLXQMNg+H@I<\ 5H@PLBED�B�M[� x<H@P[D7P�>2e I�K[x<H@P�OQJ�PLX % h = % , ) = > H@P�J'KLB�MNBEH@�<J�^�\�MNJ:� x�> ! XQK h F�XK[x]X¢� J:M[K[H_K[H_BEIu���6I<OZK[H_BEIuBEI % h > 0CX �<XQ�6I]X = ^@H@D h ! # � � � � ` �h f a.x<H@PGH�P%K[x<Xs�8<�����r����� �±- ����������� �±s*&+�¢5��r��& ��B�MYK[x]XGH@I<�6I<H_KLX��XQMNH_Bj�<H@OY\�M¬J:�6x % >���� � � ���I� � /Z ���/ �w.XQM[X%H@PzK[x]XGD7J�H�I M[X�P[�<^_KzB��tK[x<H@P�P[X�OZK[H_BEI\]

#%$�&;� ��&j( ¯ |}S;L,DnL LQ6HJe]VG*] ?(y2[;a�otGXJZBEa ��� ] d ¥ H � > f�� � ]2[�oVSrG*S0?RG%yVBEDN?Ea�k 8�LVfD_JZBtMEJZoV^sJe],BEDF?AMEJX?RuZ8�u ?Ea�?RDPqHDF? 8�S % ^ 3 JXG*SwL�Mtq0L 3 LVJ qES G*] � ?�]�? @�BEbRL2^�G*S;L 8;?EDVGXJ*GXJZBEa

Page 28: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�¤�c ����� �����#���y_[;a�opGXJXBEa�89L,D ]_JXG~L�JK]

^_B�\ )= § ?= ^_B�\ � @

?2 % � 2 [�������������� ������

��� 3 5 faYx]X��<MNB;B��z�]B+X�P[I��dKSKLX�^@^ �6PSJ�Iqi+K[x<H@I<\kJ:F�BE�<KSK[x]X7�0�<I<OZK[H_BEI � � >¢w�B#0CXQg�XQM � �

0CJ�PROZBED�� �]KLX�� 30I]BEI 8 MNH@\�B�M[BE�<P[^_i 5YF;ip�CJ�ljKLXQM dm?f 3���B�MRK[x]X \�MNJ:�6x =?>@1�J�I<� x]X�I6OZX�rB�M.J�I;iTH@PLB�MNJ��6H@J�^�\�MNJ:�6x 5¬>�<B�M � 8Z30¥ H c 5zK[x<XGJ�I<P�0CXQM.H@P� � 3 5 = § ?

= � #� #P[H@I6x(3�3 > § �5

$5+P[H�I<x\3 � �� $

5

$PNH@I<x\3 > $

5+OZBEPNx 3�$5

$ f/ I;inH�I]�rB�M¬D7J:K[H_BEITJ:F�BE�]K�K[x<H@P�H�IqKLXQ\�MNJ�^I0�BE�6^@� F�X\�MNX�J:K[^_iTJ:�<�<M[X�OQH@J:KLX���>O ����� -�5��6&;����$$P � J�O¬x�¦�� © O¬x<J�I]\�X�P�K[x]XR�6J:M[K[H_K[H@BEI ���<I6OZK[H_BEI F;i

�� = �� � >� 9 * f

��XQ� I]XGJ�I]X�0?8 �0J�I]\E^_X�� � �6J:M[K[H_K[H_BEI �0�<I<OZK[H@BEI�

� = � � >1 � � �F3��65 fe KYH@PYX�J�PLi KLB,ONx]X�O�:TK[x<J:K � �� = � �� >

aYx]X�H@�]X�J%H@PKLB��<P[X¦�§�© KLMNJ�I<PL��B�MND7J:K[H@BEI<PKLB K[�]M¬I7K[x]X�\�MNJ:�6xnH@I;KLB J%\�MNJ:�6x0.H_K[x7^@J:MN\�XYF6^_BjO�:jP 0.x<H�ONx7J:M[XOZB��6H_X�P B���F H_\R�6H_X�OZX�PB��(= > > �CX�OQJ��<PLX#%�H�P ��XQMNH_Bj�<H@O^1X�J�ONx F6H98�H@I]� I<H_KLXMNx]BED%F6�<PYO¬x<J�H@IkH@P.�6J:MNJ�^@^@X�^�KLB¢OZB��6H_X�PYB��*H_K[PLX�^_�f> E X���^@^1M[X�J:M[MNJ�I]\�XK[x]X�\�MNJ:�6x9P[B�J�PRKLB�D�B�g�X,K[x]X�P[X,ONx<J�H�I<PRJ��'&[J�OZX�IqK%KLBkX�J�ONx2B�K[x]XQM%H@I OQ^��<D��6P�1�P[BK[x<J:KK[x]X.M[X�P[�<^_K[H�I]\�MNx]BED%F6�<P K[H@^�H@I]\ROZBEI<P[H@P[K[P*B���^@J:M[\�X�MNx]BED%F6HjX�J�ONxnK[H@^_X���0.H_K[x � >OZB��6H_X�P�B�� P[D7J�^�^_XQM�g�XQMNP[H_BEI6PzB���K[x<X�D7PLX�^_g�X�PE30�tH_\E�<M[X,?�y�5¬>

w.B60��]B 0CX��]BGK[x<H�P*KLMNJ�I<P[�rB�MND,J:K[H_BEI �AzBEIqg�XQM[KzMNx<BEDGF6�6PONx6J�H@I<PCH@IqKLB PLKLMNH@I<\EPJ�P.H@IT�tH_\E�]MNX7?ZcR ^@H@�<XTK[x]XpP[KLMNH@I]\EPnJ:M[BE�<I<� 0.H_K[x]BE�<K¢OZM[X�J:K[H@I<\uI]X�0 H@I;KLXQMNPLX�OZK[H_BEI<P�> E x]X�I±JPLKLMNH@I]\¢OZM[BEP[PLX�PK[x]X H@IqKLXQM¬PLX�OZK[H_BEI B��J�I]B�K[x]XQM�6J�H_M�B��*PLKLMNH@I]\EP�16K[x]X OZM[BEP[P[H�I]\,OZB�M[M[X@8PL��BEI<�<P�KLB�Jk¦ §�© KLMNJ�I<PL��B�MND7J:K[H@BEI'BEI9K[x]X7�<I6�]XQMN^_ijH@I]\n\�MNJ:�6x�> E XnOQJ�I2P[^�H@�]XK[x]XnP[KLMNH@I]\EP J:M[BE�<I<�u�<I;K[H@^*K[x]Xn\�MNJ:�6x�^_B+B2:jP%^@H�:�X¢Jp�<I<H_BEI2B��YF6^_BjO�:jP�1tX�J�ONx�K[H@^@X��F;iu�6J:MNJ�^@^_X�^�MNx]BED%F6H�1 J�P�H@I��tH_\E�]MNXp?�y+>/aYx<J:K H@P�1 0CXTOQJ�I �]B'K[x<H@P �<I<^_X�P[P PLBED�XMNx]BED%F6�<P.O¬x<J�H@IpH@PPLXQ�6J:MNJ:KLX��p��M[BED H@K[P.KLMNJ�I6P[^@J:KLX�P�F;ikJ¢K[x<H_MN� ONx<J�H�I 0.x6H@ONx �]B;X�PI]B�K�H@I;KLXQMNPLX�OZK�X�H_K[x]XQM�F6�]KS0�x]BEPLXSMNx<BEDGF6H1x<J�g�X J,�<H ��XQM[X�I;K�OZBED,D�BEI�� J:MNJ�^@^_X�^ >Ce IK[x<H@P�OQJ�PLX�H_KRH@P���BEPNP[H_F6^_X%KLBTP[x]B#0²K[x6J:K�PNH@D��6^_i XZl]ONx<J�I<\EH@I]\ K[x]X�ONx<J�H�I<P��]B;X�PRI<B�K

Page 29: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�¤�y

=

�tH_\E�]MNX7?ZcB]

ONx<J�I]\�X%K[x]XS�6J:MNK[H_K[H_BEIT�0�<I<OZK[H_BEI�> 3 ��I]X�0zJ�iTKLBnPLXQXSK[x<[email protected]@P�KLB¢J��6�kJ�IkXZljKLMNJ�X��]\�XJ�I<�,H_KQ��P �LI]XQ\EJ:K[H_g�X� J:KCK[x]X�P[J�D7X.�6^@J�OZXPLBGK[x6J:Ki�BE�nOQJ�I¢XZl]ONx6J�I]\�X�K[x<X.K 0CB M[B#0.P�<P[H@I]\�J7P[X�A;�]X�I6OZXRB��t¦¨§2© D7B�g�X�PQ>_>_> 5

� B60 F+i/PN�]F6J��<�<H@K[H_g+H@K�i21 0.x<X�I�K[x]XQM[X�J:M[XTBEI6^_i/^�J:M[\�XTF6^_BjO�:jP�1CK[x]XT�6J:M[K[H_K[H@BEI���<I6OZK[H_BEIGOQJ�I%F�XOZBED��6�]KLX�� PLXQ�6J:M¬J:KLX�^_iRBEIGX�J�O¬x F6^_BjO�:RJ�I<�%K[x]XM[X�PN�<^_K[P�D �<^_K[H_� ^@H_X��KLB�\�XQK[x]XQM 3�0.H_K[x9J�IuXQM[M[B�M%B��z^@B#0CXQM%B�MN�<XQM5¬>T�C�<KGBEI9X�J�O¬x2F ^_B+O�:'K[x]X,\�MNJ:� x2H@P%J�6H_X�OZX�B��?=?>RJ�I<�'K[x]X7� J:M[K[H_K[H_BEI'�0�<I<OZK[H_BEI H�P�XZl+� 3+V * 3 ? @ � 3 ?#5�5�5�0.x]XQM[X V H@PRK[x<XJ:M[X�J�J�I<� * J ���<I6OZK[H_BEInB��1K[x]XSJ�I]\E^@XBEI6^_i�> ! XQKLK[H@I<\ � �

0CXSOQJ�IkOZBEI<OQ^@�6�]XH�I

K[x<J:K � )= � � -�� ��� ��� 3 5C��B�M.PLBED7X ��� >aYx<XRM[X�P[�<^_Kz��BE^@^_B60.PQ> �aYx<H@PtK[x<XQB�M[X�D x<J�P�PLBED�X.I<BEIqKLMNH@g+H@J�^]OZBEI6P�A;�<X�I<OZX�P*XQg�X�I"0.H_K[x]BE�]K :jI]B60.H@I]\ � � >

�]B�MXZl]J�D��6^_XY^@XQK � � , W F�X�K[x]X�� J:M[K[H_K[H_BEI �0�<I<OZK[H_BEI ��XQM*P[H@KLX��rB�M*K[x<X�\�MNJ:�6x�=?> 0.H@K[xX��]\�X%^_X�I<\�K[x<PS?Sg�XQMNK[H@OQJ�^@^_i J�I<� > `,x]B�MNH �QBEI;K[J�^@^_i 30J�I<� OZB�M[M[X�PL��BEI<�<H�I]\-0�X�H_\Ex;K[P �J�P J:F�B�g�X65¬> ! XQK � � � F�X,K[x<X¢�6J:MNK[H_K[H_BEIu���<I6OZK[H_BEI9��XQM P[H_KLXn�rB�M K[x]XTx]BEI]XQijOZBEDGF\�MNJ:�6x 0.H@K[x X��]\�XG^_X�I]\�K[x6PS?:>aYx]X�I

�� � � � � �= =tOZBEP^3 > OZBEPQ���63 � �> 5�5

� � ��� f/ PGJ�I]B�K[x]XQM �LJ:�<� ^@H@OQJ:K[H_BEI�nB��CK[x]X7K[x]XQB�MNi21�H_KRH@PRI<J:K[�<MNJ�^tKLB�OZBEI'&LX�OZK[�]M[X�K[x<J:K

K[x]XS�thJ8 ��XQMNOZBE^@J:K[H_BEITD�Bj�]X�^�H@P�OZMNH@K[H@OQJ�^��rB�M�0CX�H_\Ex;K[P���>*�]B�M.XZljJ�D�� ^_XRH@ITPLK[J�I<�<J:M¬���XQM¬OZBE^@J:K[H_BEIG3�� = ?#5�0CXGP[x<BE�<^@� x<J�g�X '", , 3 ?.§ '*, 5 = ��>

Page 30: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�¤�� ����� �����#���

...

...

...

�tH_\E�]MNX7?�y ]

�]B�MTXZl]J�D��6^_XpBEI�K[x]X BEI�K[x]X'x]BEI]XQijOZBED%F ^@J:KLK[H@OZXpK[x]XpOZM¬H_K[H@OQJ�^��<M[B�F6J:F6H�^@H_K�iP[x]BE�<^@�nF�X

'", = =tOZBEP � �? @ =tOZBEP � ��� ¥ f ��y�=EU�¥:c f

aYx<H@P!0zJ�PYH@I �0J�OZK.�<MNB�g�X��T��B�MYK[x]XGx<BEI]XQi+OZBED%F ^�J:KLK[H@OZXSF+i E H_XQMND7J�I dm?�¤bf >O ��� s*�65������r����! |}S;LjBRa�u kmy2[;a�otGXJZBEa : y,BED 3 SAJXo,S 3�� H 9 H * 5 = 3 : 3 ) 5 H : 3 + 5 H : 3 , 5�5]V?RGXJK]��sLV] # � % 3 S;LVa ) @ + @ , = > ,:= ?EaM # � % JK]: 3 5 = > � P[H�I�� >��� ��

P[H@I�� >��� 3 � > § 5��3 S;LVD�L = OZBEPQ���#3 > ��,:=�5 f� & ���� XS'\Y�PD.WV�. � YY�-) )�.�O �7'�� �� , 'F. �(.�, Y�V �DV�YY�-Y?'GXS'\Y � M �R �]M[�<M¬H@P[H@I]\E^@i21*K[x]X PNJ�D�XpH�PLB�MNJ��<H@J�^�H_K�i�OZBEI<�<H_K[H@BEI 0.x<H�ONx 0CB�M�:�X���PLB 0�X�^@^�H@I|K[x<XMNJ�I<�]BED OQ^@�<P[KLXQM D�Bj�]X�^CJ�^@PLB x<J�P J'P[H@D�� ^@H_��i+H@I<\nX���X�OZK�BEI/K[x]X �<H@D�XQM �6J:M[K[H_K[H@BEI���<I6OZK[H_BEI�>

Page 31: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�¤EU

! XQK % F�XSJ7F6H@�6J:M[K[H_KLX��6^@J�I6J:MY\�MNJ:�6x�> E XGX�D%F�X�� % H�PLB�MNJ��<H@J�^�^_i�> ! XQK �I3��65 =P[H@IY= B1 0.x]XQM[X GH@P*K[x]X�x<J�^_��8�J�I]\E^_XYB���K[x]X�OZB�M[M[X�PL��BEI<�<H�I]\RMNx]BED%F6�<PQ> E X��6PLXYK[x]X�PLX0�X�H_\Ex;K[PYKLBn�]XQ�6I]XSJ,�<M[B�F6J:F H@^@H_K i,D�X�J�P[�]MNXSBEIk��XQM[��X�OZK.D7J:K[O¬x<H@I]\EP.B��?3�� I<H_KLXGP[�<FB8\�MNJ:�6x<P�B��W5L% J�P�0CX�<H��7F�XQ�rB�MNX2]�K[x<X��<MNB�F6J:F6H@^@H@K�i B���J%D,J:K[ONx<H@I<\ H@P��<M[B���B�M[K[H@BEI<J�^KLB7K[x]XS�<MNB+�<�6OZKCB��tH_K[P�X��<\�X*0�X�H@\ExqK[PQ>

E X%�]XQ� I]XGJ7hSJ�P[KLX�^_XQi+I D7J:KLMNHmlkJ�P.�rBE^@^@B#0.P�]CH_� � = � 9 H@P�J�IkX��]\�XE0.H_K[xk�<�6J�^X��]\�X � < = ' < � < 1<K[x<X�Ik� 3 � H 9 5 = ; 3 ' < § � < 5 f

p* q*

w

b

�tH_\E�]MNX7?�� ]

� &j(u(u8u" � JK]j? * ?�]VG~LVu LVkEaWUW?RGXD2J 6 y,BED # ?Ea�k � a\J*G~Lj]_J*UX8u k o_BEa�aLCotG~L�M(]_[\@ 8�JZLCo_LBFy % % @� B�KLX%K[x<J:K 2 � 3 � H 9 5 2�= =tP[H@I�]>O ����� -QP efK�P[���,OZX�PzKLB7P[x<B#0 K[x<J:K���H@P�hSJ�PLKLX�^@XQi+I 8 � J:K�dm?�y2f;1]K[x<J:K�H@P�1 0CXSD%�<P[K

P[x]B#0�K[x<J:KC�rB�MCX�J�ONxn�0J�OZX�B�� % � 0.H@K[x7g�XQM[K[H@OZX�P � � H � � H fFfFf H � I H � I H@I¢OZi+OQ^@H�OYB�MN�]XQM#10�XGx6J�g�X

J:M[\g30� 3 � � H � � 5 fFfFf � 3 �I H � I 5�5 = J:M[\g3�3 §S?#5 I ��� � 3 � � H � > 5 fFfFf � 3 �

I ��� H � I 5f� 3 �I H � � 5�5 f30aYx<H@P�H@�]X�I;K[H_K�iSH�D��6^@H_X�P�K[x<J:KtK 0CB��6H@D�XQMtOZBEI]�<\E�]MNJ:K[H@BEI<PI0�x<H@ONx%BEI<^_iS�6H ��XQMtJ:M[BE�<I6�

J¢P[H�I]\E^_XG�0J�OZX�x<J�g�X�K[x<X P[J�D7X J:MN\E�<D�X�I;KH@I K[x]X XZl+� J�I<P[H_BEI B��K[x]X �]XQKLXQMND,H@I<J�I;KQ>

Page 32: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

=�¤�~ ����� �����#�����i dm?�ybf;11J�I;ipK 0CBkOZBEI]�<\E�]MNJ:K[H@BEI<PROQJ�I F�X B�F<K[J�H@I]X��p��M[BED BEI]X7J�I]B�K[x]XQMGF;ipP[�<ONx�<H@PL�6^�J�OZX�D�X�IqK[P�> 5 a1Bn�<M[B�g�X K[x<[email protected]��]X�IqK[H@K�i21�I]B�KLX%K[x6J:KH_KH@P�KLMN�]X%H@��K[x<X%��BEH@I;K[P.J:M[XM[XQ\E�<^@J:MN^@i,PL�6J�OZX�� J�^_BEI]\7J7= � 8 \�BEI\1�J�I<�TI]B�KLXSK[x<J:K�H_KzM[X�D7J�H@I<PzKLM¬�]XSH_�1i�BE��D�B�g�XBEI]XS��BEH@I;KzJ:K�J�K[H@D�X�> �

aYx]XQM[XSJ:MNXRK 0CB,D7J�H@I M[X�PN�<^_K[P�H@InK[x<H@P�P[X�OZK[H_BEI\1<J�ITXZlj�6^@H@OQH@KCOZBED�� �]K[J:K[H_BEITB�� J�I<�k� ��� >

#%$�&;� ��&j(«" |}S;L�MALtG~LVD2U�J*a�?Ea�G 89LVD ]2J*G~L�Bny � ]V?HGXJe]��sL2]^_B�\ )= ?

�[����� ��� �

?

>& 3 5 @

>^@B�\�=tP[H@I H3 S;LVD�L �I3��65 = =tP[H�IF3 3��65�5 ^ � Je]�G*S;L�a\[0U�@_LVDWBFy�bELVDVGXJZo_L2]�JKa ? y2[0aMH?EU�LVa\GI?Ru MHBEf

U6?RJKa9^�G*S9L ]_[;U JK]wBEbRLVDwG*S;LQL�Mpq L2] JKa ?Ny2[0aMH?EU�LVa\GI?RuPMHBRU6?RJKa9^ ?Ea�M & JK]zG*S;L>�BA@C?HoVS;L,bt] ;�k y_[;a�opGXJXBEa�^

& 3<;(5 = § � �� ^_B�\�=tPNH@I

$ $ f

a1BT�]X�PNOZMNH_F�XG�u����1��<MNPLK�]XQ�6I]X%�rB�MRJ�I;i OZBED��6^@XZl��6J:MNJ�D�XQKLXQM � J�I &j3r&j(u&+�����8<��°/�t��5��E��&q��&�86�18 3�°����r� -f����������� � KLB F�XJ �0�<I<OZK[H_BEInP[J:K[H@P[�rijH@I]\ : � � = : � , 3 � §� > � 5W1�H@� � � " ^_X�J��6PGJ�0zJ�i2�rM[BED�J 0.x<H@KLX21�B�M%KLB#0zJ:MN�<P JkF6^@J�O�:'g�XQM[KLXZlg1 J�I<� : � � =: � ��3 � § � > � 5W1jH_� � � " ^_X�J��<PCJ�0zJ�i¢�rMNBED J%F6^@J�O�:g1+B�M�KLB60CJ:MN�6PCJ 0.x<H@KLXYg�XQM[KLXZl�> R �<O¬xJ�I : H�PS0CX�^@^98��]XQ� I]X��TK[x]X%�<M[Bj�<�<OZKYB�� K[x]X D �<^_K[H_�6^�H_XQMNPC\�BEH@I]\ J:M[BE�<I<� J7MNx<BEDGF6�6P^_X�J��<P�F6J�O�: KLB7K[x]XGPLK[J:MNK[H@I]\�g:J�^@�]X�>

#%$�&;� ��&j( ��� L(S0?EbRL� ��� 3 9 H � � 5 = § ?

= > ; [����� ��� ���

�� � L2] � M � �� 3 : + , : �� 5 H 3 ?#5

3 S;LVD�L�G*S;L�?Ea0qRu L2] +8� ?ED�L6oVS0BE],LVaQ? 8�8�DFB98�D_JZ?RG~L,u k @ |}SAJK]�oC?Ea4@CL 3 D2JXGZG~LVa?

c > > ; � � : + 3 � 5: � 3 � 5^_B�\ � � H3 S;LVD�L � Je] ?�o,u B�],LCM�oCBRa\GIBE[;D ]_[;D_DnBE[;a�MEJ*a qNoCotu 3 G*S;L�8;?EDVG BFyPG*S;L(otJKDno,u L�� � > � 2 +8

d � § > @ � H � @ > § ��f � 3 SAJXo,S o_BEa\GI?EJ*a9](?EuKu.G*S;L�8;BEu L2](BFy : + ^ ?Ea�M 3 JXG*SNG*S9L BED2J qHJKaJKawJ*G*]�L 6HG~L,D_JZBED @

Page 33: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ =�¤�¤

O ����� -QP ! XQK � 3 9 5R�<X�I]B�KLX,K[x]X,MNH_\Ex;K 8�x<J�I<�2PNH@�]X�B��E3 ?#5¬> E X"0�H@^@^ P[x]B60 K[x<J:K� + ^ � � 3 � H 9 5 � 3 9 5 = � �� 3 � 5W1�J�I<� � 3 9 5CKLX�I6�<P�KLB �QXQM[B"0.x]X�I 9 �

>! XQK � � F�XSJ7P[D,J�^@^�^_B+B��kJ:M[BE�<I<� � > � H�I �z> E XGx<J�g�X

[+ ^ � � 3 � H 9 5 � 3 9 5 = �[L&M �

; 3 §�� > ��� @ � > ��� � � 5 � 3 9 L 5 H= § ?= > ; �[

L M �; 3 §�� > ��� @ � > ��� � � 5 [

�� �� � � � �����WYX�P � M � �� 3 : + � 5

= § ?= >

[� �� � � � ��

�� �[L&M �

3 §�� > ��� @ � > ��� � � 5 ?= > ; � � : + � 3 � 5! �= § ?

= >[

� �� � � � ��= > ; � � : � 3 � 5 �[L&M � 3 §�� > ��� @ � > ��� � � 5

3 � § � > ��� 5@3 � § � > ��� � � 5 �= § ?

= >[

� �� � � � ��= > ; � � : � 3 � 5

AC �[L&M �

?3 � § � > ��� � � 5 § ?

3 � § � > ��� 5 DG �= § ?= >

[� �� � � � ��

= > ; � � ¥ � = ¥ f

w.B60�XQg�XQMJ0.x<X�I � = � � 0CXGx<J�g�X� 3 9 L 5 = § ?

= > ; T L § L ���� > ��� § � > ��� � � Y HPLB7K[x<J:K

�[L&M �

; 3 §�� > ��� @ � > ��� � � 5 � 3 9 L 5 = ?= >

�[L&M �

L § L ��� = ? HP[H@I<OZXRK[x]XSJ�I<\E^_X�PYH@I<OZMNX�J�PLXSF+i = > J:MNBE�<I<� � � >a1B¢OZBED��6^_XQKLXSK[x<XR�<M[B+B���1]BEI]X%D%�<P[KYP[x]B#0 K[x<J:K � 3 9 5 � ¥�J�P 9 �

> �O ��� s*�65������r��� o � L(S0?RbEL

� ��� 3 9 H � 5 =?= >

T ?9 § � @ �

3 9 § � 5 Y @ � T ?2 9 § � 2 > Y H3 S;LVD�L � = : + 30¥�5 = ��� > d � � ( � � e � � > d ��� � ( � � e H ?Ea�MwG*S;L ?Ea0qRu L2]�?ED�L ?�]WJKu*u{[ ]VGXDF?HG~L�MJKa l J qH[0D�L��� @

Page 34: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

`:¥�¥ ����� �����#���

���������������������������

���������������������

���������������������������������

���������������������������������

���������������������

���������������

����������������������������������������������������

������������������������������������������������������������������������������������������������������������������������������������������������

w b

u

uu

u

u

v

v v

v12

2

3

3

4

41

θ θβ

β β

β

αα

α

α

1 21

1

2

2 3

3

4

4

4

� ���

������

��������

�tH_\E�]MNX7?�U ]

��� � /���/Z���)���$� ���- �]B�MRJ7�6I<H@KLXG\�MNJ:�6x % 0�X �]XQ�6I]X%K[x]X ���6�:(28 3����E&+����&q��&+�:(u�0��86��� 1<B�M��]XQKLXQMND7H�8I<J�IqK s*&;�¢5��r��& 16�]XQK ��� B��tJ�IkB���XQM¬J:KLB�M � ] � � � � � KLB7F�X

�]XQK ����� ���= 2 �<XQK � 2 � � � � � H0.x]XQM[X 2 % 2 H@P�K[x]XSI+�<D%F�XQM�B��tg�XQM[K[H�OZX�PYB���% >�]B�MSJ�IpB���XQMNJ:KLB�M � BEI'J¢� I<H_KLXG\�M¬J:�6x\1��]XQK � H@PJ¢���<I6OZK[H_BEI�B��K[x]X�D7J:KLMNHml

X�IqKLMNH@X�P � 3 ; H 7 5�0.x<H@O¬x|H@P7^�H@I]X�J:M,H@I|X�J�ONx±X�IqKLM[i PLXQ� J:MNJ:KLX�^_i�> efI¡�6J:M[K[H@OQ�6^@J:M��rB�MJ�I X��<\�X � = ; 7B1�J�P�Jn���<I6OZK[H_BEIkB��*K[x<X D7J:KLM¬HmlkX�IqKLMNi � 3 ; H 7 5 0CX x<J�g�X��]XQK � =� @ � 3 ; H 7 5W1�0.x]XQMNX H@P�3 §G?#5 > ( L K[H@D7X�P�K[x<X��]XQKLXQMND,H@I<J�I;K1B��<K[x]XzD7H@I<B�M1B�F<K[J�H@I<X��F;inM[X�D�B�g+H@I<\�M[B#0 ; J�I<�TOZBE^@�6D7I 7]>CaYx<J:KYH@P�1

�H30�]XQK � 5�H3 � 3 ; H 7 5�5= � ��� 3 7 H ; 5 ���]XQK � HB�M �H30^_B�\C�<XQK � 5� � 3 ; H 7 5

= � ��� 3 7 H ; 5 f 3 =�5

Page 35: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ `:¥]?

R �]�<��BEPLX�K[x6J:K % H@PS��XQMNH@B+�<H�O��<I<�]XQMSKLM¬J�I<P[^@J:KLX�P%F;i9Jp^@J:KLK[H@OZX��.> ! XQK % h =% , ) �S1+K[x]X�6I<H_KLX.\�MNJ:�6x�0�x<H@ONx¢H@P*K[x<X A+�]B�K[H_X�I;KB�� % F;i ) ��> � B60�H@�g0CX�P[�<D 3 =�5�rB�M�J�^@^�� 8 KLMNJ�I6P[^@J:KLX�PB�� X��]\�X ; 7TH@I % h 1�J�I<� K[x]X�I'�<H_gjH@�]XRF�B�K[x PNH@�]X�PYF+i 2 % h 2 1�H@Ki+H_X�^��<P �H30^_B�\��]XQK � � 5� � 3 ; H 7 5

= ?2 % � 2 � ��� 3 7 H ; 5 H 3 `�5

0.x]XQM[X � 3 ; H 7 5CH@PCI]B#0 K[x]X�OZBED7D�BEI40CX�H_\Ex;K�B��1J�^�^6KLMNJ�I6P[^@J:KLX�PCB���X��]\�X ; 7 1+K[x6J:KzH�P�1K[x]X^_XQ��K 8�x<J�I<�%P[H@�]X*H@P�K[x]X�O¬x<J�I]\�X�H�IS^_B�\��<XQK � � 0�x]X�IRK[x]X 0CX�H_\Ex;K1B��<J�^@^EKLMNJ�I<P[^@J:KLX�PB�� ; 7nO¬x<J�I]\�X�PQ> � B�KLXGK[x6J:KYK[x<H@PzX�A+�<J:K[H_BEI H@P�H@I<�]XQ��X�I<�<X�IqKCB�� ) >

e K H@P I]B#0 JpP[x]B�M[K OZBED��6�]K[J:K[H@BEIuKLB OZBED�� �]KLX¢K[x<Xn�]XQKLXQM¬D7H@I<J�I;KGB��.� �rM[BEDK[x]XSXZl]J�OZKY�rB�M¬D B�� � ��� >_>_>_>�PLXQX4dm?�`bf

�� � & ' ��-� �$����2 /g��� / ���L���10� aYx]X�PLXQK*B���H@PLB�MNJ��6H@J�^+X�DGF�X��<�<H�I]\EP�e R �-3 %E5W1 0.x<X�I �6J:MNJ�D�XQKLMNH�PLX�� F;i K[x]X�MNx<BEDGF6�6PJ�I]\E^_X�P�1qH�P OZBEIqg�XZl�>*aYx<H@P���BE^@^_B60.PtF�X�OQJ��<PLX�J�I7H@PLB�M¬J��<H@J�^+X�DGF�X��<�6H@I]\�H@Pt�]XQKLXQMND7H@I<X���rM[BEDbJ,P[XQK�B��*M¬x]BEDGF �<PYJ�I]\E^_X�P�F+ikK[x]X%^�H@I]X�J:M�OZBEI6�<H_K[H_BEI<PYK[x<J:K�K[x<X P[�6D J:M[BE�<I6�X�J�ONxkg�XQMNKLXZl�D%�<P[K�F�X > >aYx]XQM[XH@PCJ%�<I6H�A;�<XE3���BEPNP[H_F6^_i �]XQ\�X�I]XQM¬J:KLX65CH@PLB�MNJ��<H�J�^]X�DGF�X��<�6H@I]\GD7J�l]H@D7H ��H@I]\ 1]F�X�OQJ��<P[X H�PYPLKLMNH@OZK[^@i¢OZBEI<OQJ�g�X%BEITe R �-3 %*5 f3���BEH@I;K�0.H@K[x �^8 ­2> R ONx6^_X�IB:�XQM5 / �q��4 � �q8<4 �6J:K[xkH@ITJ�� ^@J�I<J:MY\�MNJ:� xkH@P�J��6J:K[x0.x<H@O¬x2K[�]M¬I<P%D,J�ljH@D,J�^@^_i ^_XQ��K J:K7J g�XQM[KLXZl(1K[x]X�I�D7J�l]H@D7J�^@^@ipM¬H_\ExqK J:K K[x]XTI]XZljKg�XQM[KLXZlg1+K[x]X�I7D,J�ljH@D,J�^@^_i%^@XQ�rK�1+J�I<�7PLBRBEI�> aYx6J:KH@P�1qH@K*^_X�J�g�X�P��rMNBED JRg�XQM[KLXZl,J�^_BEI<\J�IGX��]\�X�0.x<H�ONxGH�P1J��'&[J�OZX�IqK H@I%OZi+OQ^�H@O B�MN�]XQM�KLBK[x]XX��]\�XzH_K�X�IqKLXQM[X�� BEI\1�J�^_KLXQM¬J�IqK[H@I<\KLBTK[x]X�MNH_\Ex;KRJ�I<� KLBTK[x]X7^_XQ��KQ> / �6^�J�I<J:M�\�MNJ:�6x x<J�PSJ�I H�PLB�MNJ��<H@J�^�X�D%F�X��<�<H@I]\ H_�J�I<�TBEI<^_inH_��K[x]XR�rBE^�^_B#0.H�I]\ K 0CB¢OZBEI<�<H@K[H_BEI<P�J:M[XSP[J:K[H�PL�<X��Z30�tH_\E�]MNX7?�~�5W]

?:> � B ��H_\^84��J:\,� J:K[xkOZM[BEP[PLX�P�H_K[PLX�^_�=+> � B7K 0CBD��H_\^84��J:\¢�6J:K[x<PYOZMNBEP[PYD�B�M[XGK[x<J�ITBEI<OZX�>

�tH_\E�]M[X7?�~ ]�� �_ %��_`_����.�F�P� � � ¤Z�C�~�2�p� ��_ aC�~�_£���¤

Page 36: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

`:¥E= ����� �����#���e KH@PI]B�K�x<J:M¬��KLB PLXQX 0.xqi K[x]X�P[X OZBEI<�6H_K[H_BEI<P�J:M[X I<X�OZX�P[P[J:M[ig]YJ ��H_\^84��J:\ �6J:K[x

OZB�M[M[X�PL��BEI<�<P*KLB%JSM¬x]BEDGF �<P*ONx6J�H@I,H@I,J�I,H@PLB�M¬J��<H@J�^]X�D%F�X��<�<H@I]\B1;J�I<�7P[�<O¬x¢JGONx<J�H@[email protected]�BEI<B�KLBEI]X%J�I<�pPLBnOQJ�I<I]B�K�OZMNBEP[P�H@K[PLX�^_�f>za�0�BnP[�<ONx ONx<J�H�I<P.OQJ�I<I<B�KOZM[BEPNP�D�B�MNXK[x<J�ITBEI<OZXG�6�]XRKLB7K[x]XSB�M¬H_X�IqK[J:K[H@BEITB���K[x<X�H_M.OZBED7D�BEI�MNx]BED%F6H >

���� � + /Z�-� 2���� � � / � 2 32�+ � / �$�-�a1BSJ�I7H�PLB�MNJ��<H@J�^�^_iSX�D%F�X��6�]X�� \�MNJ:�6x % 0�XYOQJ�I¢J�P[PLBjOQH@J:KLX.J�I7H@�<X�J�^j��BE^@i+x]X��<M[BEI%H�I� J�P��rBE^�^_B#0.P�> R �]�6��BEP[X�%|H�P1X�D%F�X��<�]X��SBEI%K[x]X�; = �6^�J�I]X 0.x<H�ONxGH�P�K[x]XCF�BE�<I<�<J:M[i

J:K

B�� K[x]X �]�<��XQM.x<J�^_��8�PL�6J�OZX D�Bj�]X�^�B�� � >�� XQM[K[H@OZX�PB�� � J:MNX%g�XQM[K[H�OZX�P�B��L% <

30OQH_MNOQ^_XGOZX�I;KLXQMNPW5.J�I<�TX��<\�X�P.B�� � J:M[XG\�XQBj�]X�P[[email protected]<H@O¬xT�<M[B'&LX�OZK.g�XQMNK[H@OQJ�^@^_i KLB,K[x<XX��]\�X�PYB���% < >

aYx]X�g�BE^@�<D�X�B�� � ��XQM����<I6�<J�D�X�I;K[J�^6�]BED7J�H�I¢OQJ�I¢F�X�PLXQX�InKLB F�X.K[x]XPNJ�D�XJ�PK[x]XSX�I;KLM[B��;i ��XQM��0�<I<�<J�D�X�I;K[J�^��]BED7J�H�I B���K[x<XG�<H@D�XQM�D�Bj�]X�^�BEI % >aYx<J:KYH�P�1

?

>� BE^ 3�� 5 = ?

>[� ^ ��� � � & 3 5 = ^@B�\ § [

� ^ ��� � � {*M#3��b5+^_B�\ �F3��65 f­kB�MNXQB�g�XQM�K[x]XRKLXQMND[

� ^ ��� � � {*M#3��b5+^_B�\ �I3��65 = [� ^ ��� � �

>^@B�\C=tP[H@I�

OQJ�I F�XGJ�PNPLB+OQH�J:KLX�� 0�H_K[xkK[x]X D�X�J�I'OQ�]M[g:J:K[�]M[X%B�� �D1 0�x<H@ONx H@P�F;ik�]XQ� I<H_K[H_BEITK[x<XP[�<D B�g�XQM�K[x<XGX��]\�X�P�B�� K[x]X%�6H@x]X��]MNJ�^�J�I]\E^@XGK[H@D�X�P.K[x]Xe30I<B�MND7J�^@H �QX��g5zxqi+��XQM[F�BE^@H@OX��]\�XG^_X�I]\�K[x�>� ��&+5Q����� �$P AzJ�IRBEI]XOZBEI6PLKLMN�<OZK�J�K[H�^@H@I]\zH@IRPLBED�X�OQJ�I]BEI<H@OQJ�^ 3�X�I;KLM[B��+i�8 �6M[X�PLXQM[gjH@I]\ 50CJ�i �6P[H@I]\ K[x]XS\�XQBj�]X�P[H�O �<B#0¨H@I<P[H@�<X �

Page 37: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

������������ ����������������������������! "�#��$ `:¥E`

� � R�� '���-, ��

dm?f%WS>�CJ�ljKLXQM�1 � l]J�OZK[^_i�PLBE^_g�X��¨D�B+�<X�^@PkH@I P[K[J:K[H@PLK[H@OQJ�^%D�X�ONx<J�I6H@OQPQ> / OQJ��]X�D7H�O{*M[X�PNPQ>

dd=6f%WS>��C�<M[KLBEI\1�WG>�{�X�D7J�I;K[^_X21 ! BjOQJ�^O¬x<J:MNJ�OZKLXQMNH@P[K[H@OQP�1�X�IqKLM[B��+i9J�I<�u^@H@D7H_K�K[x]XQB^8M[X�D7P���B�M�PL� J�I<I<H@I]\�KLMNXQX�P J�I<�%�<BED7H@I]BYK[H�^@H@I]\[email protected]�I<PL��XQM�8�H@D7��X��6J�I<OZX�P�1 � a�a @cjDnBA@ @ !� 1�I]B]> `B3 ?�¤�¤�`�5%?�`�=�¤68N?�`EU+?:>

dd`6f%wG> AzBEx<I\1 � > � ^�:+H_X�P�1 �<>j{M[B��<�F1 ! BjOQJ�^�PLK[J:K[H@P[K[H@OQPz�rB�MzMNJ�I6�]BEDª�]BED7H@I]B K[H@^@H�I]\EPB��tK[x]X / �QKLX�O �<H@J�D7BEI<��>�� [ ;HL�O4?RG*S @ � @ � v 3 ?�¤�¤���5W1�I]B]>�?^1�?�?�U�� ?����+>

d c^f%wG>;AzBEx<I\1�WS>�hRX�I;i�BEI\1 �<>�{M[B��<�F1 / g:J:MNH@J:K[H@BEI<J�^+�<MNH@I<OQH@�6^_X*��B�M �]BED7H@I]BK[H@^@H�I]\EPQ>� @ � UWL,D @ OQ?RG*S @ g�Bto @ q� 3 =:¥�¥]?#5W1�I]B]>6= 1�=�¤EU��+`�cq�+>

ddy6f%WS> �]>z�����,I\1Y{�B�KLX�I;K[H@J�^�K[x]XQB�M[i|BEI�J/MNx]BED%F6H@Ok^�J:KLK[H@OZX�> � @ vxBRUW@,JKa?RGIBED2JX?Eu|}S;L�BRD_k v 3 ?�¤���~�5�=�y�~��+=EU:=+>

dd�6f � > � ^�:jH_X�P�1j� >qhS�]��XQM[F�XQM[\B1q­2> ! J:MNPLX�I\1 �]>q{M[B��<�F1 / ^_KLXQM¬I<J:K[H@I]\^8�P[H@\EI7D7J:KLMNH@OZX�PJ�I<�k�]BED,H@I]B K[H@^@H@I<\EPQ>je¬> � @ � u q Lp@,DF?RJXo vxBEU�@,JKa @ 3 ?�¤�¤�=�5W1�I]B]>�= 11?�?�?�� ?�`�=+>

d�U#f � >j��MNH@D7D�XQKLK�1+aYx]X.P[KLB+O¬x<J�PLK[H@O�MNJ�I<�]BEDD8�OQ^@�<PLKLXQM*�6M[B+OZX�PNPJ�I<�,K[x]X.�<I<H�A;�]X�I<X�P[PB��tMNJ�I<�]BED-8�OQ^��<PLKLXQMYD�X�J�P[�<M[X�PQ> � a\a @ cjDnBA@�?A@ @ !6o 3 ?�¤�¤�y�5W1�I<B]>]cB11?Zcq�j?�� ?�yj?Q¥j>

dd~6f%{�> E >*hSJ�P[KLX�^_XQi+IF1aYx]XTPLK[J:K[H@P[K[H@OQP B����<H�D�XQMNP%BEI J ^@J:KLK[H@OZX�> e >ta.x]XnI+�<D%F�XQMB���<[email protected]:M[MNJ�I]\�X�D7X�IqK[P�BEIpJ"A+�<J��]MNJ:K[H�OG^@J:KLK[H@OZX21 csS k�]2JXo_? !�" 3 ?�¤��j?#5W1?�=:¥E¤��?�=�=�y+>

dd¤6f%{�> E ><hSJ�P[KLX�^_XQi+IF1 �MNJ:�6xnK[x]XQB�M[i J�I<� OZM[ijPLK[J�^��6x;i+PNH@OQP�1�?�¤��EU7�MNJ:� xna.x]XQB�M[iJ�I<�kaYx<XQB�M[XQK[H@OQJ�^�{�xqijP[H@OQP��6��>jcq`�� ?�?Q¥ 1 / OQJ��]X�D7H�OR{*M[X�P[P�1 ! BEI<�]BEI

dm?Q¥bf WS>�hRX�Iqi�BEIF1 ! B+OQJ�^�PLK[J:K[H@PLK[H@OQPGB��Y^@J:KLK[H@OZX,�6H@D�XQMNP�1 � a�a @ � a�]2G @��<@ cdBEJ*a�o_?ED!TLV^cjDnBA@ @ L,G�g\GI?RG @ o6o 3 ?�¤�¤EU25W1�y�¤j?8 �j?�~+>

dm?�?f WS>%hRX�Iqi�BEIF1 ACBEI]��B�MND7J�^,H@Iqg:J:MNH@J�I6OZX B�� �<BED7H@I]B�K[H@^�H@I]\B1 � a�a @ c�DFBA@ @ !<�3 =:¥�¥�¥�5¬>

dm?�=6f WS>;hRX�Iqi�BEIF1;�BED7H@I]BEP�J�I<��K[x]XR��J��<P[PNH@J�I���M[XQXY�<X�^@�F1 � a�a @ c�DFB @ @ !<£ 3 =:¥�¥]?#5¬>dm?�`6f WS>�hRX�Iqi�BEIF1�aYx<X ! J:�6^@J�OQH�J�ISJ�I<�S��H_MNJ�O*B���XQM¬J:KLB�MNP�BEISOZM¬H_K[H@OQJ�^��6^�J�I<J:M�\�MNJ:�6x<P�1

�<M[XQ�<M¬H@IqKQ>

dm?Zc^f WS> hRX�Iqi�BEIF1"�]>6{M[B��<�F1<� > E H@^�PLBEI\1<a�M[XQX�P.J�I<� ­�J:K[O¬x<H@I]\EP�1 u @ � @ vxBEU�@VJ*a @" 3 =:¥�¥�¥�5W1�WYX�P[X�J:MNONx��6J:��XQM�=�y 1 `�c��<��>

Page 38: efI;g XQMNPLXGhGJ PLKLX ^ XQijIkD7J:KLMNHmln>S>S ...users.ictp.it/~pub_off/lectures/lns017/Kenyon/Kenyon.pdf&+46 8

`:¥:c ����� �����#���dm?�y6f�� >�hS�]��XQM[F�XQM[\B1 / I'XZl+�6^@B�MNJ:K[H_BEIpB��K[x]X ��XQMND7J�I<X�IqK 8��]XQKLXQM¬D7H@I<J�I;KD7XQK[x]B+��>

u L�opGXDFBEa @ � @ vxBEU�@VJ*a @ v 3 ?�¤�¤�~�5W1�I]B]>�?^1�W.X�PLX�J:MNO¬x¡{tJ:��XQM�cq� 1C`�c �<��>�3�X�^_X�O8KLM[BEI<H@O#5

dm?��6f�A>�­�XQMNOQJ:K�1���H@PNOZM[XQKLX7W.H_X�D,J�I<I'P[�]M[�0J�OZX�PRJ�I<�'K[x]X�e P[H@I]\ D�Bj�]X�^ > vxBEU�U�[;a @OQ?RG*S @ csSAk�] @ !� ;� 3 =:¥�¥]?#5W1@?�U�U#8 =j?��j>

dm?�U#f w%>�a�X�D���XQMN^_XQi21EACBED%F6H@I<J:KLB�MNH�OQP�]1{*MNB+OZXQX��<H�I]\EP�B��<K[x<X��CMNH_K[H@PNx ACBED%F6H@I<J:KLB�M¬H@J�^ACBEI]��XQM[X�I<OZX�?�¤EU:` 1 >}BEa�MHBEa O4?RG*S @ gBto @ >%L�opGX[0D�L���BRG~L2]WgL,D_JIL2]����R� 1?3 ?�¤EU�c 5=:¥E=68 =:¥

dm?�~6f ­2>j�tH@PNx]XQM�1]wG>]a�X�D���XQMN^_XQi21jaYx]XR�<H@D7XQM��<MNB�F6^_X�D H@I PLK[J:K[H@PLK[H@OQJ�^�D7X�ONx<J�I<H�OQP �J�IkXZl]J�OZK.M[X�PN�<^_KQ> csSAJ*uO4?tq @ ¯ 3 ?�¤��j?#5W1t?Q¥E�j?8N?Q¥E��`+>

dm?�¤6f��]> E H_XQMND7J�I\1*�CBEI<�u��XQMNOZBE^@J:K[H_BEI/BEI�x]BEI]XQijOZBED%F�J�I6�uKLM¬H@J�I]\E�<^@J:M ^@J:KLK[H@OZX�PQ>� MRb @ J*a � 8-8�u @ c�DFB @�?A@ @ +o 3 ?�¤�~j?#5W1�I]B]>6= 1 =�¤�~��+`j?�`+>