[2015/03/12]

The stochastic higher spin vertex model introduced in my paper with Ivan Corwin ([arXiv:1502.07374 [math.PR]][ivan6v]) generalizes the stochastic six vertex model considered by Borodin, Corwin, and Gorin, arXiv:1407.6729 [math.PR]. Here are some simulations related to this model.

Feel free to use these pictures to illustrate your research in talks and papers, with attribution (CC BY-SA 4.0). Some of the images are available in very high resolution upon request.

[2015/02/25]

We introduce a four-parameter family of interacting particle systems on the line which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain Markov dualities.

[2015/02/18]

I’ve implemented the Glauber dynamics to (approximately) sample uniformly random lozenge tilings
of polygons of Gelfand-Tsetlin type. These polygons are called sawtooth domains by J. Novak. This paper by **B. Laslier** and **F.L. Toninelli** establishes rate of convergence of the Glauber dynamics to the uniformly random lozenge tiling.

This talk describes results on spectral theory for q-Hahn zero-range process, ASEP, six-vertex model, and q-TASEP. Based on [14] and [17].

[2014/07/31]

We develop spectral theory for the $q$-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result which implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system.

[2014/02/10]

I have collected a number of facts about Markov chains that were discussed in lectures 5-9 in the graduate probability course in Spring 2014.

[2014/02/07]

Asymptotic representation theory of general linear groups $GL(n,q)$ over a finite field leads to studying probability measures $\rho$ on the group $U$ of all infinite uni-uppertriangular matrices over $F_q$, with the condition that $\rho$ is invariant under conjugations by arbitrary infinite matrices.

[2014/01/10]

A colleague addressed a question to me and others, and after discussing how arXiv orders submitted preprints we’ve concluded that the rules seem to be like this.

[2014/01/10]

I “grew up” in mathematical sense learning about Fock spaces and
some nice things one can do with them.
There always was a mysterious physical term
“**second quantization**”
attached to the subject
(wikipedia). Recently I was reading
Felix Berezin’s lecture notes from 1966-67;
they are called “*Lectures on Statistical Physics*”,
in English: translated from the Russian and edited by Dimitry Leites),
in Russian: MCCME 2008.
The full English text can be readily found.

Chapter 25 of these lecture notes contains a clear and historic description of second quantization, in a unified way for Bosons and Fermions. Let me briefly record how this is done.

[2014/01/05]

Looking for an answer to a colleague’s question, I was
reading a 1962 seminal paper
by Freeman J. Dyson:
*“A Brownian-Motion Model for the Eigenvalues of a Random Matrix”* published in
J. Math. Phys. 3, 1191 (1962).

The question was, how the original proof of Dyson’s result about his Brownian motion of eigenvalues of random matrices was carried out? I will try to reproduce Dyson’s arguments. His physics paper was missing some of the computations; I will also not go to all technical details, but rather work out finite cases at an elementary level.

Arguments similar to the original Dyson’s ones also appear (in a more advanced form) in this post by T. Tao. He deals with the proof of Theorem 2 (Dyson’s Brownian motion), but does not mention the proof Theorem 1 (density of eigenvalues) in that post.