q-deformed RSKs and Random Polymers

Konstantin Matveev

(joint work with Leo Petrov, University of Virginia)

Harvard University

May 25th, 2015

Robinson-Schensted-Knuth correspondence

n× t matrices A of nonnegative integers.Ordered pairs (P,Q) of semistandard Young tableaux

of the same shape - P with entries from {1, 2, . . . , n},Q with entries from {1, 2, . . . , t}

bijection

P = ← ω1 ← ω2 · · · ← ωt

A11 A12

A21

A1t

A2t

An1 AntAn2

A22

· · ·

· · ·

· · ·

· · ·

ωj := 1︸︷︷︸A1j

2︸︷︷︸A2j

. . . n︸︷︷︸Anj

consecutive row insertions

interlacing array of integers

· View t as time – array evolves.

· Each row insertion – first someinteraction among the rightmostparticles, which then spreads

· Random entries – random dynamics.

· More precisely, random dynamics of

throughout the array.

the rightmost particles, which spreadsto the array by deterministic rule.

Column insertion

consecutive column insertions

· Each column insertion – first someinteraction among the lefttmostparticles, which then spreads

· Random entries – random dynamics.

· More precisely, random dynamics of

throughout the array.

the leftmost particles, which spreadsto the array by deterministic rule.

Dual operation – column insertion

ω∗j := n︸︷︷︸

A1j

n− 1︸ ︷︷ ︸A2j

. . . 1︸︷︷︸Anj

P ∗ = (ω∗t → · · ·ω∗

2 → ω∗1 →)

Dual tableaux

Overview

RSK row/column insertion → deterministic lifting of 1D interacting system to 2D interacting system

For some particular distributions of entries –integrable (precise formulas for observables)1D random interacting particle system.

q-deformed integrable 1D random

q-deformed RSK row/column insertion → random lifting of

interacting particle systems

of 1D integrable particle system to 2D random interacting

system. This lifting depends on the system.

Evolving with time partition functionsof log-gamma random polymers

Evolving α-Whittaker process

q → 1 limit

q → 1 limit

Geometric RSK column/row insertion evolution forgamma/inverse-gamma distributed entries.

TASEP systems

TASEP (continuous time): The k-th

Particles that try to jump, but are blocked by right neighbor, stay put

particle tries to jump to the rightindependently of others at rate ak.

xk−1xk

q-TASEP (continuous time): The k-th particle triesto jump to the right independently of others at rateak(1− qgap(k)), gap(k) := xk−1 − xk − 1, gap(1) :=∞.

Borodin-Corwin, 2011

q = 0

Geometric q-TASEP (discrete time): The 1-st particle jumps by m with probabilityαm

1 (α1;q)∞(q;q)m

.

The k-th particle (k ≥ 2) jumps by m with probabilityαm

k (αk;q)gap(k)−m(q;q)gap(k)

(q;q)m(q;q)gap(k)−m.

Bernoulli q-TASEP(discrete time): Particles update their positions sequentially from right to left

A) If the k − 1-st particle has jumped, then the k-th particle jumps with probability αk

1+αk.

B) If the k − 1-st particle hasn’t jumped, then the k-th particle jumps with probability αk(1−qgap(k))1+αk

.

Borodin-Corwin, 2013

Take αk := εak (Bernoulli) or εak(1− q)(Geometric), make one step of dynamics

In discrete time at each step of dynamicsall particles move.

xk−1xk

x1x2x3xn . . .

and can jump by at most 1. The first particle jumps with probability α1

1+α1.

xk−1 has jumpedxkxk−1 hasn’t jumpedxk

Jumps with probability αk

1+αk

Jumps with probability αk(1−qgap(k))1+αk

x1x2x3xn · · ·

occur in time ε and let ε→ 0.

Initial condition: xk = −k.

PushTASEP systems

PushTASEP (continuous time): The k-th

Particles jump, even if blocked by right neighbor, and push this neighbor.

particle jumps to the right at rate ak andpossibly initiates an avalanche of pushings.

q-PushTASEP (continuous time): The k-th particlejumps to the right at rate ak, then it pushes the k + 1-stparticle with probability qgap(k), and so on...

Borodin-Petrov, 2013

Geometric q-PushTASEP (discrete time): Particles update their positions sequentially from left to right.Given the move of the k-th particle by m, the k + 1-st particle jumps due to the push by p with probability

Bernoulli q-PushTASEP(discrete time): Particles update their positions sequentially from left to right

A) If the k-th particle hasn’t jumped, then the k + 1-st particle jumps with probability αk+1

1+αk+1.

B) If the k-th particle has jumped, then the k + 1-st particle jumps with probability αk+1+qgap(k)

1+αk+1.

M. - Petrov, 2015

xk+1xk

and can jump by at most 1. The first particle jumps with probability α1

1+α1.

xk+1xk hasn’t jumped.

Jumps with probability αk

1+αk

xk xk+1 xk+2 xk+3 · · · xk+2qgap(k) qgap(k+1) if k + 1-st jumped. · · ·

q = 0

xnx3x2x1 · · ·

qgap(k)p(qgap(k);q−1)m−p(q−1;q−1)m

(q−1;q−1)p(q−1;q−1)m−p. and then jumps by s with probability

αsk+1(αk+1;q)∞

(q;q)s.

xk xk+1 spm

xk+1xk has jumped.

Jumps with probability αk+1+qgap(k)

1+αk+1.

Take αk := εak (Bernoulli) or εak(1− q)(Geometric), make one step of dynamicsoccur in time ε and let ε→ 0.

Initial condition: xk = k

gap(k) := xk+1 − xk − 1

Semistandard Young Tableaux with Interlacing particle arrays on n levels

65322

2 3 55

5

4

4

2

32

1 5311144322

4334455

Row insertion P ← x ”Pull” wave Column insertion x→ P ”Push” wave

1 5311144322

4334455

1 5111143322

4334455

← 1

4

65322

2 3 55

5

4

4

2

32

+1

+1

+1

+1 1 5311144322

4334455

1 4311143322

4334455

3→

65322

2 3 55

5

4

4

2

32

+1

+1

+1

5+1

λ1 = (4)λ2 = (4, 2)

λ3 = (5, 3, 2)

λ4 = (5, 5, 3, 2)

λ5 = (6, 5, 3, 2, 2)

RSK row and column insertions

entries from {1, 2, . . . , n}

Column insertion of Several ”Push” waves

1 5311144322

4334455

1 3311143322

4334455

65322

2 3 55

5

4

4

2

32

4

433→

4

+1 +1

+3

+31 5311144322

4334455

65322

2 3 55

5

4

4

2

32

Row insertion of

1 2211143322

4334455

← 224

45

4+2

+1

+1

+1

+1

+2

+1 +1

Several ”Pull” waves

5

a word P ← ω a word ω → P

RSK random dynamics on interlacing arrays

At time t a column of independent random entriesA1t, A2t, . . . , Ant is attached to the table and theword ωt is row inserted in the array.

At time t a column of independent random entriesA1t, A2t, . . . , Ant is attached to the table and theword ω∗t is column inserted in the array.

(fix a1, a2, . . . , an > 0)

Bernoulli distribution:P (Ait = 0) = 1

1+aiβt,

P (Ait = 1) = aiβt1+aiβt

Bernoulli distribution:P (Ait = 0) = 1

1+an+1−iβt,

P (Ait = 1) =an+1−iβt

1+an+1−iβt

Geometric distribution:P (Ait = k) = (1− aiαt)(aiαt)kfor k = 0, 1, 2, . . .

Geometric distribution:for k = 0, 1, 2, . . . P (Ait = k) =

Distribution on the array after t steps starting from zeroinitial conditions is the Schur process given by P(λ) =(∏n

i=1

∏t

j=11

1+βjai

)(∏n

i=1a|λi|−|λi−1|i

)Sλn(β1, . . . , βt)

Distribution on the array after t steps starting from zeroinitial conditions is the Schur process given by P(λ) =(∏n

i=1

∏t

j=1(1− αjai)

)(∏n

i=1a|λi|−|λi−1|i

)Sλn(α1, . . . , αt)

Sλ(β1, . . . , βt) is the image of the Schur function under a homomorphism from the ring of symmetric functions to real

numbers given by∑∞

m=1xkm → (−1)k−1

∑t

j=1βkj .

row Bernoullidynamics

row geometricdynamics

column Bernoullidynamics

column geometricdynamics

(1− an+1−iαt)(an+1−iαt)k

Schur function is Sλ =∑T− semistandard tableaux x

number of 1′s1 xnumber of 2′s

2 xnumber of 3′s3 · · ·

RSK dynamics

q-deformations

We seek q-deformations of these four dynamics.

row Bernoulli column Bernoulli

row geometric column geometric

Qqrow[β] Qq

col[β]

Qqrow[α] Qq

col[α]

?

? ?

?

What should the q-deformations of these dynamics be?

”Come, listen, my men, while I tell you again

The five unmistakable marks

By which you may know, wheresoever you go,

The warranted genuine Snarks.Lewis Carroll, ”The Hunting of the Snark”.

1). (Distribution) The distribution of the array at a particular moment of time should be

2). (Update) Each dynamics should be of sequential update type .

3). (Degenerations) For q = 0 these dynamics should be classical RSK dynamics. In the

4). (Projection) The leftmost or the rightmost particles of the array should evolve in a

5). (Naturalness) The dynamics should evolve through ”contiguous interactions” and each

a q-Whittaker process instead of Schur process.

continuous limit we should recover constructions of Borodin-Petrov (row insertion case)

and O’Connell-Pei (column insertion case).

marginally markovian manner according to some 1D integrable dynamics.

progressive wave of movements should travel all the way to the upper level.

Distribution

skew Schur functions Sλ/µ → skew q-Whittaker functions Pλ/µ and Qλ/µ

Schur processes → q-Whittaker processes:

Qλ(β1, . . . , βt) is the image of the q-Whittaker function under a homomorphism from the ring of symmetric functions to real

(∏ni=1

∏tj=1

11+βjai

)Sλn(a1, . . . , an)Sλn(β1, . . . , βt) or

λn

λn−1

λ1

(∏ni=1

∏tj=1(1− αjai)

)Sλn(a1, . . . , an)Sλn(α1, . . . , αt)

Stochastic Links in the Schur case

P(λn → λn−1) =Sλn−1 (a1,...,an−1)a

|λn|−|λn−1|n

Sλn (a1,...,an)

P(λ2 → λ1) =Sλ1 (a1)a

|λ2|−|λ1|2

Sλ2 (a1,a2)

P(λn−1 → λn−2) =Sλn−2 (a1,...,an−2)a

|λn−1|−|λn−2|n−1

Sλn−1 (a1,...,an−1)

...

Top floor measure in the Schur case P(λn) = (∏ni=1

∏tj=1

11+βjai

)Pλn(a1, . . . , an)Qλn(β1, . . . , βt) or(∏n

i=1

∏tj=1(αjai; q)∞

)Pλn(a1, . . . , an)Qλn(α1, . . . , αt)

Top floor measure in the q-Whittaker case P(λn) =

Stochastic Links in the q-Whittaker case

P(λn → λn−1) =Pλn−1 (a1,...,an−1)Pλn/λn−1 (an)

Pλn (a1,...,an)

P(λ2 → λ1) =Pλ1 (a1)Pλ2/λ1 (a2)

Pλ2 (a1,a2)

P(λn−1 → λn−2) =Pλn−2 (a1,...,an−2)Pλn/λn−1 (an−1)

Pλn−1 (a1,...,an−1)

numbers given by∑∞m=1 x

km → (−1)k−1(1− qk)

∑tj=1 β

kj .

UpdateSequential update type dynamics:

λn

λn−1

λ1

...

νn

νn−1

ν1

...

Probability of such transition at time step t− 1→ t is of the form

U1(λ1 → ν1 | )U2(λ2 → ν2 | λ1 → ν1) · · ·Un(λn → νn | λn−1 → νn−1).

∑νkUk(λk → νk | λk−1 → νk−1) = 1, (λk−1, λk, νk−1 are fixed),

update on the

1st level

update on the

2nd level

update on the

nth level

Ber: For Qqrow[β] and Qqcol

[β] Bernoulli dynamics

Geo: For Qqrow[α] and Qqcol

[α] geometric dynamics

∑λk−1 Uk(λk → νk | λk−1 → νk−1)(βtak)|λ

k|−|νk|−|λk−1|+|νk−1|

ψλk/λk−1ψ′νk−1/λk−1 = 11+βtak

ψνk/νk−1ψ′νk/λk , (λk, νk, νk−1 are fixed).

∑λk−1 Uk(λk → νk | λk−1 → νk−1)(αtak)|λ

k|−|νk|−|λk−1|+|νk−1|

ψλk/λk−1φνk−1/λk−1 = (αtak; q)∞ψνk/νk−1φνk/λk , (λk, νk, νk−1 are fixed).

Equations from the condition on distributions:

ψλ/µ :=∏`(µ)i=1

(λi − λi+1

λi − µi

)q

ϕλ/µ := 1(q;q)λ1−µ1

∏`(µ)i=1

(µi − µi+1

µi − λi+1

)q

ψ′λ/µ :=∏i≥1:λi=µi,λi+1=µi+1+1(1− qµi−µi+1)

Systems of linear equations:we are interested only innonnegative (!) solutions.

Uk( | ) ≥ 0.

Projections

q - row Bernoulli Qqrow[β] q - column Bernoulli Qq

col[β]

q - row geometric Qqrow[α] q - column geometric Qq

col[α]

Rightmost particles (after shift xk := λk1 + k) Leftmost particles (after shift xk := λkk − k)

Bernoulli q-PushTASEP Bernoulli q-TASEP

geometric q-PushTASEP geometric q-TASEP

Continuous time limits I

Borodin-Petrov, 2014:Row dynamics Qqrow[α] andQq

row[β] → construction of

Rightmost particle at the j-th level jumps independently

from the others with rate aj . Each move triggers move

of one particle on the upper level

λj−1k

λjkλjk+1

1−qλjk−λj−1

k

1−qλj−1k−1

−λj−1k

qλjk−λj−1

k −qλj−1k−1

−λj−1k

1−qλj−1k−1

−λj−1k

In α, β → 0 limit (time rescaled)

Continuous time limits II

O’Connell-Pei, 2013:Column dynamics Qqcol

[α] and Qqcol

[β] → construction of

Particle λjk jumps independently from the

aj(1− qλj−1k−1

−λjk)q

∑j−1

i=kλj−1i

−λji+1 if 1 < k < j,

aj(1− qλj−1j−1

−λjj ) if k = j,

ajq∑j−1

i=1λj−1i

−λji+1 if k = 1.

Each move triggers move of one particle on

. . . . . .

λj−1k

λjkλjm λj1

1−qλj−1k−1

−λjk

1−qλj−1k−1

−λj−1k

(qλj−1k−1

−λjk−q

λj−1k−1

−λj−1k )(1−q

λj−1m−1

−λjm )q

∑k−2

i=mλj−1i

−λji+1

1−qλj−1k−1

−λj−1k

(qλj−1k−1

−λjk−q

λj−1k−1

−λj−1k )q

∑k−2

i=1λj−1i

−λji+1

1−qλj−1k−1

−λj−1k

λj−1k−1

In α, β → 0 limit (time rescaled)

others with rate

the upper level:

q - Bernoulli Row Construction Qqrow[β]

”island splitting”.

+1 +1 +1+1+1+1+1

+1 +1 +1 +1 +1 +1 +1

. . .

. . .

. . .

. . .

Given update λ = λj−1 → νj−1 = ν on the j − 1-st level, to determine update λ = λj → νj = ν on the j-th level first

choose Birth of a new impulse (with probabilityβtaj

1+βtaj): |νj | − |λj |+ |λj−1| − |νj−1| = 1 and the rightmost

Then idependent splitting of islands

Let fi := 1−qλi−λi

1−qλi−1−λi, gi := 1− qλi−λi .

λkλs−1λsλm

λkλk+1λs−1λsλs+1λmλm+1

fk

. . .

. . .

. . .

. . .

λkλs−1λsλm

λkλk+1λs−1λsλs+1λmλm+1

(1− fk)(1− gk+1) · · · (1− gs−1)gs

. . .

. . .

. . .

. . .

λkλs−1λsλm

λkλk+1λs−1λsλs+1λmλm+1

(1− fk)(1− gk+1) · · · (1− gm−1)(1− gm)

Probability

Except in the case of Birth the rightmost particle

on the j-th level moves and the island containing

particle on the j-th level moves or No Birth (with probability 11+βtaj

): |νj | − |λj |+ |λj−1| − |νj−1| = 0.

the rightmost particle on the j − 1-st level (if it

exists) splits the first way.

q - Bernoulli Column Construction Qq

col[β]”particle detachment”.

Birth of a new impulse (with probabilityβtaj

1+βtaj): |νj | − |λj |+ |λj−1| − |νj−1| = 1 or No Birth (with probability

11+βtaj

): |νj | − |λj |+ |λj−1| − |νj−1| = 0. Then idependent detachment of rightmost impulse from each island.

f ′k

(1− f ′k)(1− g′k−1) · · · (1− g′s+1)g′s

. . .

. . .

λrλkλk+1λm

λsλkλk+1λm

(1− f ′k)(1− g′k−1) · · · (1− g′r+3)(1− g′r+2)

Probability (r might be 0)Let f ′i := 1−qλi−1−λi

1−qλi−1−λi,

. . .

. . .

. . .

. . .

. . .

. . .

λrλkλk+1λm

λsλkλk+1λm

. . .

. . .

. . .

. . .

. . .

. . .

λrλkλk+1λm

λsλkλk+1λm

. . .

. . .

. . .

. . .

λr+1

λr+1

λr+1

In case of Birth also another particle

λr

λsλj+1

. . .

. . . . . .λr+1

leftmost island

λr

λs+1λj

. . .

. . . . . .λr+1

leftmost island

λr

λsλj

. . .

. . . . . .λr+1+1

leftmost island

g′j(1− g′j)(1− g′j+1) · · · (1− g′s+1)g′s

(1− g′j)(1− g′j+1) · · · (1− g′r+2)

Probability

moves, the leftmost if r = j − 1,if r < j − 1 - according to:

g′i := 1− qλi−1−λi .

Complementation

U ′j(λ→ ν | λ→ ν) := (ajβt)−2(|λ|−|ν|−|λ|+|ν|)−1Uj([S − λ]→ [S + 1− ν] | [S − λ]→ [S + 1− ν])

for S large enough.

for Qqcol

[β] for Qqrow[β]

λ

[S − λ]

[S − λ] := S − λj ≥ S − λj−1 ≥ · · · ≥ S − λ1

. . . λ1λ2λj−1λj S

S − λjS − λj−1S − λ2S − λ1

Reversedirection

q - Geometric Row Construction Qqrow[α]

”impulse allocation among neighbors”

q-deformed beta distribution on {0, 1, . . . , y} –

Given update λ = λj−1 → νj−1 = ν on the j − 1-st level, determine update λ = λj → νj = ν on the j-thlevel first independently for each 1 ≤ k ≤ j − 1

λk

λkλk+1

νk

dνk − λk − d

with probability ηq−1,qλk−λk ,qλk−1−λk (d | νk − λk)

Then the leftmost particle on the j-th level makes additional

jump ` with probability ηq,ajαt,0(` | ∞) = (ajαt; q)∞(ajαt)

`

(q;q)`

0 + 2

2 + 3 8 12 + 4 19 + 5 29 + 4

6 + 1 9 + 2 15 + 5 25 + 5 35 + 1 + 3

+2 +1 +2 +2 +3 +2 +3 +1 additional jump

with P(s) = ηq,µ,ν(s | y) := µs (ν/µ;q)s(µ;q)y−s(ν;q)y

(q;q)y(q;q)s(q;q)y−s

q - Geometric Column Construction Qqrow[α]

”immediate vs. reserve pushing”

Given update λ = λj−1 → νj−1 = ν on the j − 1-st level, determine update λ = λj → νj = ν on the j-th level firstby successively updating positions of particles from left to right

λk νk

λkvoluntary movement x immediate push y push from the Stabilization Fund z

ηq,ajαtqV ,0(x | λk−1 − λk), V is the sum of

λk−1

ηq−1,qνk−λk ,qλk−1−λk (λk−1 − λk − x− y | λk−1 − λk − x)

ηq−1,qS ,0(λk−1 − λk − x− y − z | λk−1 − λk − x− y),

voluntary movements of particles λj , . . . , λk+1.

where S is the current value of the Stabilization Fund.

After such step the Stabilization Fund is updated

S → S + νk − λk − y − z.

For k = 1 just take y = ν1 − λ1, z - current value

of the Stabilization Fund.

Limit q → 1q = e−ε, aj = e−θjε, αt = βt = e−θtε;

θj , θt > 0, ε→ 0+.

Geometric q-PushTASEP

Rescale:

xj(t)− j =: (t+ j − 1)ε−1 log ε−1 + ε−1 logRj1(t, ε)

Log-gamma polymer

ajt ∼ Gamma−1(θj + θt)

Gamma distribution X ∼ Gamma(θ) – P (X ∈ dx) = 1Γ(θ)x

θ−1e−xdx; P (X−1 ∈ dx) = 1Γ(θ)x

−θ−1e−1/xdx.

Geometric q-PushTASEPx1x2x3xn · · ·

Rescale (for t ≥ j):xj(t) + j =: (t− j + 1)ε−1 log ε−1 − ε−1 logLj1(t, ε)

xnx3x2x1 · · ·

Rj1(t) :=∑π:(1,1)→(t,j)

∏(s,i)∈π a

is

Seppalainen, 2012

π – up/right lattice paths

Strict-weak polymer

ajt ∼ Gamma(θj + θt)

Lj1(t) :=∑π:(0,1)→(t,j)

∏(s−1,i)→(s,i)∈π a

is

Corwin - Seppalainen-Shen; O’Connell-Ortmann, 2014

π – up-and-right/right lattice paths

Rj1(t, ε)→ Rj1(t)

M.-Petrov, 2015

Lj1(t, ε)→ Lj1(t)

CSS, 2014

Array limit q → 1rj,k(t, ε) - postion of the k-th particle from the righton the j-th level after t steps of Qq

row[α] dynamics.

(t+ j − 2k + 1)ε−1 log ε−1 + ε−1 log Rjk(t, ε)

For t ≥ k

Rjk(t) - sum over k-tuples of nonintersecting paths

from (1, 1), . . . , (1, k) to (t, j − k + 1), . . . , (t, j) in

the setting of log-gamma polymer.

Rjk(t) := Rj

k(t)/Rjk−1(t)

Then

Rjk(t, ε)→ Rj

k(t)

rj,k(t, ε) =:

`j,k(t, ε) - postion of the k-th particle from the lefton the j-th level after t steps of Qq

col[α] dynamics.

(t− j + 2k − 1)ε−1 log ε−1 − ε−1 log Ljk(t, ε)

For t ≥ j − k + 1

Ljk(t) - sum over k-tuples of nonintersecting paths

from (0, 1), . . . , (0, k) to (t, j − k + 1), . . . , (t, j) in

the setting of strict-weak polymer.

Ljk(t) := Lj

k(t)/Ljk−1(t)

Then

Ljk(t, ε)→ Lj

k(t)

`j,k(t, ε) =:

Geometric / (de)tropical RSK row insertion

Geometric row inserion

Rjk(t)

(detropicalization via (max, +) → (+, x))

Geometric / (de)tropical column insertion

Ljk(t)

Geometric column insertion(detropicalization via (min, +) → (+, x))