# Yang-Baxter equation for the asymmetric simple exclusion process (ASEP)

On page 14 of Craig Tracy's slides on ASEP, it states that the $n$-particle boundary condition can be reduced to the 2-particle boundary condition due to the fact that the $S$-matrix satisfies the Yang-Baxter equation.

I assume the $S$-matrix is the Yang-Yang $S$-matrix on page 16:

$$S_{\alpha\beta} = - {p + q \xi_\alpha \xi_\beta - \xi_\alpha \over p + q \xi_\alpha \xi_\beta - \xi_\beta}$$

My question is, what is the Yang-Baxter equation written in terms of these $S$-matrices?

I can find on the Internet introductions to the Yang-Baxter equations in the language of quantum spin chains (with relevant examples of XXX and XXZ chains), but I can't spot in them any equations involving the $S$-matrix defined above. Thanks in advance.

• There is a dictionary of parameters from the spin model. Nov 18, 2016 at 17:56
• @AHusain Where can I find this dictionary? Nov 22, 2016 at 14:49
• Crampe Ragoucy Simon. You can follow the sources if you want original source. Nov 22, 2016 at 18:24
• when you say that the $S$ matrix satisfies the Yang-Baxter equation, can you be more specific? Which Yang-Baxter equation do you mean: the classical Yang-Baxter eq., the quantum Yang-Baxter equation, some other version ? Nov 27, 2016 at 23:11
• @KonstantinosKanakoglou I don't know for sure but I guess it's the quantum one, because the (quantum) Yang-Baxter Equation $R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u)$ seems to pop up in notes about the XXZ chains, and the XXZ chain is the ASEP. But like AHusain said there are two different languages the probabilistic one and the quantum spin chain one and I feel like it's just a matter of translation. Nov 28, 2016 at 2:30

The $S$-matrix you write can be written as $\frac{x_\alpha-Q x_\beta}{x_\alpha-x_\beta}$, where $x_{\alpha,\beta}^{}$ are some fractional linear transformations of $\xi_{\alpha,\beta}^{}$, and $Q$ is the ratio of ASEP's $p$ and $q$. If you go one step up to the stochastic six vertex model (which ASEP is a limit of), then the Yang-Baxter equation for the stochastic six vertex model's $R$-matrix (or in some literature it's referred to as $L$-matrix) involves this $S$-matrix.
Maybe you can find p. 13 in https://arxiv.org/pdf/1601.05770v1.pdf helpful, or for example p. 6 on the slides here http://newton.kias.re.kr/~namgyu/index.html/CA16/slides/Borodin.pdf has a more graphical interpretation (both refs deal with higher spin version of the stochastic six vertex model, but the Yang-Baxter equation works equally well for the six vertex case, with the same $S$-matrix). The $S$-matrix is the vertex weights in the vertex with diagonal edges.
• Thanks for your answer. What is $x_\alpha$ in terms of $\xi_\alpha$ explicitly? What is the Yang-Baxter equation for the stochastic six vertex model involving the $S$-matrix? I fail to find it on page 13 of arXiv:1601.05770 or page 6 of the slides. The vertex weights in the vertex with diagonal edges in the above references are ${u - s q^g \over 1 - s u}$. How is it related to the $S$-matrix? Nov 22, 2016 at 14:46
• @Y.Pei first, $x=(\xi - 1)/(Q \xi - 1)$. Second, what do you mean by YB equation for the S matrix? I think of something like SLL = LLS, where L are vertex weights like $(u-sQ^g)/(1-su)$ and S is that S matrix you're asking about. For the stochastic six vertex you need to set $s=1/\sqrt{Q}$, and then $u_{1,2}$ are spectral parameters of two vertices one above another in the slides, they play the role of $x_{1,2}$. Nov 22, 2016 at 15:37