Stochasticization of Solutions to the Yang–Baxter Equation

Abstract

In this paper we introduce a procedure that, given a solution to the Yang-Baxter equation as input, produces a stochastic (or Markovian) solution to (a possibly dynamical version of) the Yang-Baxter equation. We then apply this "stochasticization procedure" to obtain three new, stochastic solutions to several different forms of the Yang-Baxter equation. The first is a stochastic, elliptic solution to the dynamical Yang-Baxter equation; the second is a stochastic, higher rank solution to the dynamical Yang-Baxter equation; and the third is a stochastic solution to a dynamical variant of the tetrahedron equation.

Mapping TASEP back in time

“…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.…”

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

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

“…. .. 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) ≺ . .…”

“…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 .…”