Stochasticization of Solutions to the Yang–Baxter Equation

Borodin, Alexei; Bufetov, Alexander I.

doi:10.1007/s00023-019-00799-y

Cited by 11 publications

(28 citation statements)

References 56 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Denote by C the (countable) space of configurations on Z which differ from the step configuration by finitely many TASEP jumps. 1 Consider the continuous-time Markov chain on C which evolves as follows. At each hole there is an independent exponential clock whose rate is equal to the number of particles to the right of this hole.…”

Section: The Backwards Dynamicsmentioning

confidence: 99%

“…We will only need the particular case of ascending Schur processes. These are probability measures on interlacing arrays λ (1) ≺ λ (2) ≺ . .…”

Section: Schur Processesmentioning

confidence: 99%

“…By the Kolmogorov extension theorem, a measure on S is uniquely determined by a collection of compatible joint distributions of {λ (1)…”

Section: Remark 28mentioning

confidence: 99%

“…. .. A probability distribution on S is called c-Gibbs if for any N , given λ (N ) = λ, the conditional distribution of the bottom part λ (1) ≺ . .…”

Section: C-gibbs Measuresmentioning

confidence: 99%

“…Let us first describe the maps for j = 1 (the simplest nontrivial case) to illustrate their structure and properties. We use the shorthand notation λ (2) = (λ 1 , λ 2 ) and λ (1) = κ 1 . The interlacing means that λ 2 ≤ κ 1 ≤ λ 1 .…”

Section: First Levelmentioning

confidence: 99%

See 4 more Smart Citations

Mapping TASEP back in time

Petrov

Saenz

2021

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

We obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

show abstract

Section: The Backwards Dynamicsmentioning

confidence: 99%

“…We will only need the particular case of ascending Schur processes. These are probability measures on interlacing arrays λ (1) ≺ λ (2) ≺ . .…”

Section: Schur Processesmentioning

confidence: 99%

“…By the Kolmogorov extension theorem, a measure on S is uniquely determined by a collection of compatible joint distributions of {λ (1)…”

Section: Remark 28mentioning

confidence: 99%

“…. .. A probability distribution on S is called c-Gibbs if for any N , given λ (N ) = λ, the conditional distribution of the bottom part λ (1) ≺ . .…”

Section: C-gibbs Measuresmentioning

confidence: 99%

Section: First Levelmentioning

confidence: 99%

See 3 more Smart Citations

Mapping TASEP back in time

Petrov

Saenz

2021

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

show abstract

Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model

Aggarwal

2019

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

In this paper we consider the stochastic six-vertex model on a cylinder with arbitrary initial data. First, we show that it exhibits a limit shape in the thermodynamic limit, whose density profile is given by the entropy solution to an explicit, non-linear conservation law that was predicted by Gwa-Spohn in 1992 and by Reshetikhin-Sridhar in 2018. Then, we show that the local statistics of this model around any continuity point of its limit shape are given by an infinite-volume, translation-invariant Gibbs measure of the appropriate slope.

show abstract

Stochastic PDE Limit of the Six Vertex Model

Corwin

Ghosal

Shen

et al. 2020

Commun. Math. Phys.

View full text Add to dashboard Cite

We study a stochastic PDE limit of the height function of the dynamic asymmetric simple exclusion process (dynamic ASEP). A degeneration of the stochastic Interaction Round-a-Face (IRF) model of [Bor17], dynamic ASEP has a jump parameter q ∈ (0, 1) and a dynamical parameter α > 0. It degenerates to the standard ASEP height function when α goes to 0 or ∞. We consider very weakly asymmetric scaling, i.e., for ε tending to zero we set q = e −ε and look at fluctuations, space and time in the scales ε −1 , ε −2 and ε −4 . We show that under such scaling the height function of the dynamic ASEP converges to the solution of the space-time Ornstein-Uhlenbeck (OU) process. We also introduce the dynamic ASEP on a ring with generalized rate functions. Under the very weakly asymmetric scaling, we show that the dynamic ASEP (with generalized jump rates) on a ring also converges to the solution of the space-time OU process on [0, 1] with periodic boundary conditions.

show abstract

Stochasticization of Solutions to the Yang–Baxter Equation

Cited by 11 publications

References 56 publications

Mapping TASEP back in time

Mapping TASEP back in time

Limit Shapes and Local Statistics for the Stochastic Six-Vertex Model

Stochastic PDE Limit of the Six Vertex Model

Contact Info

Product

Resources

About