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.
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.
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.
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.
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.
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.
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.
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.
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.