Blog entries, page 2

A flowchart for Macdonald processes and some of their specializations and limits (from talks of the authors on the subject)

5 years of Macdonald Processes


Today marks a 5-year anniversary of the paper “Macdonald Processes” by A. Borodin and I. Corwin. It was posted on the arXiv on November 18, 2011 ( and was subsequently published at Probability Theory and Related Fields (2014), Volume 158, Issue 1, pp 225–400. As of this day, Google Scholar counts 178 citations to this paper.

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Sauron's eye

Pictures and movies of random lozenge tilings


In Spring 2015 and Summer 2016 I used the Python and Mathematica as briefly described here to produce a number of pictures and “movies” of random lozenge tilings. Some of these pictures even appeared at an art exhibition at Harvard’s Radcliffe Institute accompanying Alexei Borodin’s fellowship.

These pictures and “movies” of random tilings are collected in this post.

Dear colleagues:

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.

Pictures and movies »

Moscow in December 2015

[travel] 2015 travel


25-29 • Paris • Inhomogeneous Random Systems conference

All travel in 2015 »

Fluctuations of the stochastic six vertex model

Simulations of stochastic higher spin vertex model


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.

Dear colleagues:

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.

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Uniformly random tiling of a 9-gon

Implementation of Glauber dynamics simulation of random lozenge tilings


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.

See here the many results of the simulations.

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Facts about Markov chains


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.

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[tech] What time of the day should I submit a preprint to arXiv?


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.

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Second quantization


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.

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Wigner's semicirle law

How Dyson obtained his Brownian motion in 1962


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.

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