Mapping TASEP back in time
Leonid Petrov
University of Virginia and IITP

joint work with Axel Saenz
arXiv:1907.09155 [math.PR]• TASEP and ASEP (with particles moving in two directions) were introduced in1969-1970, independently in biology [C. MacDonald, J. Gibbs, and A. Pipkin ‘69]
and probability [Spitzer ’70]

• In these 50 years, we understood a lot about TASEP and related systems (in theKardar-Parisi-Zhang universality class), including limit shapes and fluctuations
with general initial data

• Yet new asymptotic results are added every year (KPZ fixed point, Airy sheet,directed landscape, …). Let us give one basic example of asymptotics…

Parabola limit shape [Rost 1981]
Fluctuations [Johansson 2000]

Parabola limit shape [Rost 1981]
Fluctuations [Johansson 2000]

Definition. Backwards Hammersley process• The Markov chain lives on left-packed configurations
x1 > x2 > x3 > …

• Each hole has an independent exponential clock with rate equal to

the number m of particles to its right, ℙ(wait > s) = e −m⋅s, s > 0.

• When the clock at a hole rings, the leftmost of the particles that areto the right of the hole instantaneously jumps into this hole

• Because total rate of jump is proportional to the size of the gap, thisis a discrete space inhomogeneous version of the Hammersley
process [Hammersley ’72], [Aldous-Diaconis ‘95]

Running TASEP back in time
Theorem [P.-Saenz]. Let μt be the distribution of the TASEP (with stepIC) at time t. Let Lτ be the backwards Hammersley Markov semigroup.
Then μt Lτ = μt⋅e −τ, i.e.,∑x⃗μt( x ⃗)Lτ( x ,⃗ y ⃗) = μt⋅e−τ( y ⃗).

Note. Theorem doesnot apply to multitime
distributions

Simulation joint with Haoyu Li(UVA undergraduate)Running TASEP back in time
Theorem [P.-Saenz]. Let μt be the distribution of the TASEP (with stepIC) at time t. Let Lτ be the backwards Hammersley Markov semigroup.
Then μt Lτ = μt⋅e −τ, i.e.,∑x⃗μt( x ⃗)Lτ( x ,⃗ y ⃗) = μt⋅e−τ( y ⃗).

Note. Theorem does
not apply to multitime
distributions

Simulation joint with Haoyu Li(UVA undergraduate)Equilibrium dynamics on TASEPCorollary. Run in parallel:• The usual TASEP;• The backwards Hammersley process slowed down by a factor of t
The combined process preserves the TASEP distribution μt.

Simulation by Haoyu Li(UVA undergraduate)Equilibrium dynamics on TASEPCorollary. Run in parallel:• The usual TASEP;• The backwards Hammersley process slowed down by a factor of t
The combined process preserves the TASEP distribution μt.

Simulation by Haoyu Li(UVA undergraduate)Strategy of the proof• TASEP is mapped to Schur measures (e.g. via RobinsonSchensted-Knuth correspondence, known since 1990s)

model structure (Uq( 𝔰𝔩2 ̂) model with q = 0 and infinitevertical spin)• Schur measures in a “bosonic” interpretation have a vertex• Vertex weights satisfy the Yang-Baxter equation• The Yang-Baxter equation can be turned into a stochasticmap• This stochastic map leads to the backwards Hammersley
process and the theorem

Schur vertex model and Yang-Baxter (RLL) identity
A higher spin six vertex model with q = 0 and infinite vertical spin
Vertex weights

1uu

Cross vertex weights

Yang-Baxter equation
1=Sum over g1, j1, j2 of the left-hand sideis equal to the sum over g2, j′1, j′2 of theright-hand side

Schur vertex model and Yang-Baxter (RLL) identity
A higher spin six vertex model with q = 0 and infinite vertical spin
Vertex weights

1uu

Cross vertex weights

Yang-Baxter equation
1=Sum over g1, j1, j2 of the left-hand sideis equal to the sum over g2, j′1, j′2 of theright-hand side

Example:Schur symmetric polynomials
Let λ be a partition with N parts,
λ = (λ1 ≥ λ2 ≥ … ≥ λN ≥ 0), λi ∈ ℤ.
The Schur polynomial sλ(u1, …, uN ) is defined asthe partition function of the vertex model:

Vertex weights

1uu1

Another formula:λj+N−j N
det[xi]i,j=1sλ(u1, …, uN ) =det[xiN−j]Ni,j=1

The partition function sλ(u1, …, uN ) is symmetric thanks to the Yang-Baxter equation

Bijectivisation of the Yang-Baxter equation
Let A, B be finite sets and∑w(a) =a∈A∑w(b)(with positive terms)b∈B

A bijectivisation (coupling) of this identity is a family of transition probabilitiesp(a → b) and p′(b → a), satisfying

w(a)p(a → b) = w(b)p′(b → a)
for all a ∈ A, b ∈ B.
If all probabilities are equal to 0 or 1 and | A | = | B | , thenthis is a usual bijection.

Bijectivisation of the Yang-Baxter equation
Let A, B be finite sets and∑w(a) =a∈A∑w(b)(with positive terms)b∈B

A bijectivisation (coupling) of this identity is a family of transition probabilitiesp(a → b) and p′(b → a), satisfying
Example: 1 + 3 = 2 + 2w(a)p(a → b) = w(b)p′(b → a)

for all a ∈ A, b ∈ B.

(maximallydependent)

If all probabilities are equal to 0 or 1 and | A | = | B | , thenthis is a usual bijection.

(independent)Bijectivisation of the Yang-Baxter equation
Let A, B be finite sets and∑w(a) =a∈A∑w(b)(with positive terms)b∈B

A bijectivisation (coupling) of this identity is a family of transition probabilitiesp(a → b) and p′(b → a), satisfying
Example: 1 + 3 = 2 + 2

w(a)p(a → b) = w(b)p′(b → a)(maximallydependent)

for all a ∈ A, b ∈ B.
If all probabilities are equal to 0 or 1 and | A | = | B | , thenthis is a usual bijection.

(independent)

For the Yang-Baxter equation:leads toBijectivisation in the vertex modelWith u > v > 0, denote the distribution of κ byPu,v(ϰ ∣ λ, μ) ∝ u|ϰ|−|μ| |λ|−|ϰ|v(where | ϰ | = ϰ1 + … + ϰM).

u
∝
(v)

|ϰ|

Bijectivisation in the vertex modelWith u > v > 0, denote the distribution of κ byPu,v(ϰ ∣ λ, μ) ∝ u|ϰ|−|μ| |λ|−|ϰ|v(where | ϰ | = ϰ1 + … + ϰM).

u∝
(v)

|ϰ|

Lemma. For u > v, the bijectivised YBE maps
the measure Pu,v to Pv,u and acts as follows,
where α = v/u:Same map with right jumps maps Pv,u back to Pu,v(so, where you jump depends on the order of u, v).

Remark. Action of the symmetric group SN
Let u1, …, uN > 0 be distinct spectral
parameters. With each σ ∈ SN we

associate a measure 𝕄σ on configurations
with top condition λ and spectral
parameters uσ(1), …, uσ(N).

The Markov operators or Lα( j) and Rα( j) withappropriate α act by transpositions on σ.

Remark. Action of the symmetric group SN
Let u1, …, uN > 0 be distinct spectral
parameters. With each σ ∈ SN we

associate a measure 𝕄σ on configurations
with top condition λ and spectral
parameters uσ(1), …, uσ(N).

The Markov operators or Lα( j) and Rα( j) withappropriate α act by transpositions on σ.

In particular, the sequence (1,q, q 2, …, q N−1),
0 < q < 1, can be mapped to its reversal(q N−1, …, q 2, q,1) by only applying the left
jump operators L ( j), totally ( ) of themN2(lozenge tilings are in bijection
with vertex configurations)
Simulation joint with Edith Zhang(UVA undergraduate)q = 0.7

Remark. Action of the symmetric group SN
Let u1, …, uN > 0 be distinct spectral
parameters. With each σ ∈ SN we

associate a measure 𝕄σ on configurations
with top condition λ and spectral
parameters uσ(1), …, uσ(N).

The Markov operators or Lα( j) and Rα( j) withappropriate α act by transpositions on σ.

In particular, the sequence (1,q, q 2, …, q N−1),
0 < q < 1, can be mapped to its reversal(q N−1, …, q 2, q,1) by only applying the left
jump operators L ( j), totally ( ) of themN2(lozenge tilings are in bijection
with vertex configurations)
Simulation joint with Edith Zhang(UVA undergraduate)q = 0.7

Limit shapes of q vol lozenge tilings: [Cohn-Kenyon-Propp ’00], [Kenyon-Okounkov ’05]

q = 0.95

Simulation joint with Edith Zhang(UVA undergraduate)Limit shapes of q vol lozenge tilings: [Cohn-Kenyon-Propp ’00], [Kenyon-Okounkov ’05]

q = 0.95

Simulation joint with Edith Zhang(UVA undergraduate)TASEP at the edge of a vertex model
Via Robinson-Schensted-Knuth (since 1990s),[O’Connell ’03], also [Borodin-Ferrari ’08]

Vertex weights

1uu

1Prob[λ (N)(t) = λ] = e −(u1+…+uN )t sλ(u1, …, uN )sλ(ρt),
tt
where sλ(ρt) := lim sλ( , …, ) (K times)
KK
K→+∞
Then yi(t) coming from this vertex model are identified with particle
coordinates of TASEP with speeds ui at time t.

( j)Edge of a vertex model and L . Finishing the proof
Take the spectral parameters (speedsin TASEP) to be 1,q, q 2, …, where0 < q < 1.
Apply the left jumps in the vertex
model in this order.
Then the measure sλ(1,q, q 2, …)sλ(ρt)
turns into sλ(q, q 2, q 3, …)sλ(ρt), andsλ(q, q 2, q 3, …)sλ(ρt) = q |λ|sλ(1,q, q 2, …)sλ(ρt) = sλ(1,q, q 2, …)sλ(ρq⋅t)
which uses homogeneity of the Schur polynomials.
We thus proved:( j)Edge of a vertex model and L . Finishing the proofTake the spectral parameters (speedsin TASEP) to be 1,q, q 2, …, where0 < q < 1.
Apply the left jumps in the vertex
model in this order.
Then the measure sλ(1,q, q 2, …)sλ(ρt)
turns into sλ(q, q 2, q 3, …)sλ(ρt), andsλ(q, q 2, q 3, …)sλ(ρt) = q |λ|sλ(1,q, q 2, …)sλ(ρt) = sλ(1,q, q 2, …)sλ(ρq⋅t)
which uses homogeneity of the Schur polynomials.
We thus proved:Theorem. Denote by μt(q) the measure of the TASEP with q-speeds, and 𝕃q the(q)combination of the left jump operators as above. Then μt(q)𝕃q = μq⋅t.

Applying 𝕃q many times and taking Poisson-like limit q → 1 leads to the main theorem.

Remark. Back in time in a two-dimensional model
The Borodin-Ferrari Anisotropic KPZ dynamics (arXiv:0804.3035 [math-ph]) on twodimensional interlacing arrays also has a reversal.

Forward dynamics is defined via push-block rules:• Each vertical arrow has an independent exponential clockwith rate 1. When the clock rings, the arrow attempts tomove to the right by 1 (in its horizontal line)• If the jumping arrow is blocked from below, there is nojump• If the arrow’s jump violates interlacing with above, pushingis forced

Remark. Back in time in a two-dimensional model

Simulation by Patrik Ferrari https://wt.iam.uni-bonn.de/ferrari/research/jsanimationakpz/

Remark. Back in time in a two-dimensional model

Simulation by Patrik Ferrari https://wt.iam.uni-bonn.de/ferrari/research/jsanimationakpz/

Remark. Back in time in a two-dimensional model
The reversal dynamics is defined as follows. Each possible(*) hole at each level k has
an independent exponential clock of rate k. When the clock rings, the hole attracts theclosest of its right neighbors.

Like for the TASEP reversal (backwards Hammersley), some care is needed to define
the dynamics as dependence propagates “from infinity” (“possible holes” depend on
the lower and the upper levels); and there are infinitely many jumps in finite time.
For push-block initial data this is possible, and we get the time reversal Mt𝕃τ = Mt⋅e −τConsequences for TASEPTheorem. μt Lτ = μt⋅e −τ• This is a new structural result for TASEP with step initial data
• Likely characterizes the distribution μt by a stationary dynamics (onehas to prove convergence to the distribution μt)• Leads to new identities for expectations under μt, for example:
Here ht is the number of particles to the right of zero, and G( ⋅ ) is
an arbitrary function.

• The presence of “two times” in TASEP raises questions about fluctuation
exponents. For example, if we run TASEP for time t and stationarydynamics for time s, what are the combined fluctuation exponents?

• Limit of stationary dynamics to the top Airy line?
• Other initial data?

Summary. More questions than answers• Found a new interesting property of TASEP• Characterization of nonequilibrium distributions by means of stationarydynamics• How general is this eﬀect?

- Seems that it applies (in one way or another) to most integrable
particle systems in the KPZ universality class: stochastic six-vertex
model, random polymers, edge of random matrices…

• New asymptotic questions - do some known good tools work?
• Time reversal applies to some two-dimensional models, too• Any other stochastic applications of YBE? Yes, at least [Grimmett-Manolescu 2014] in critical percolation; and domino shuﬄing for Aztec

diamond [Elkies-Kuperberg-Larsen-Propp 1992] are essentially based on
bijectivisations of YBE