All entries (with papers and talks), page 4

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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O'Connell-Yor directed random polymer

[talk] Integrable Probability: Random Polymers, Random Tilings, and Interacting Particle Systems

One of the versions of my “job talk” in Fall 2013 describing many facets of integrable probability. It is supposed to be accessible. The talk is based on [15] and on the previous two talks. More details can also be found in arXiv:1106.1596 [math.PR] by Corwin and arXiv:1212.3351 [math.PR] by Borodin and Gorin.

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Large scale and long time behavior of a jumping interacting particle system

[paper] Integrable probability: From representation theory to Macdonald processes


These are lecture notes for a mini-course given at the Cornell Probability Summer School in July 2013. Topics include lozenge tilings of polygons and their representation theoretic interpretation, the $(q,t)$-deformation of those leading to the Macdonald processes, nearest neighbor dynamics on Macdonald processes, their limit to semi-discrete Brownian polymers, and large time asymptotic analysis of polymer’s partition function.

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[paper] Spectral theory for the q-Boson particle system


We develop spectral theory for the generator of the $q$-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system.

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[paper] The q-PushASEP: A New Integrable Model for Traffic in 1+1 Dimension


We introduce a new interacting (stochastic) particle system $q$-PushASEP which interpolates between the q-TASEP introduced by Borodin and Corwin (see arXiv:1111.4408, and also arXiv:1207.5035; arXiv:1305.2972; arXiv:1212.6716) and the $q$-PushTASEP introduced recently by Borodin and Petrov (arXiv:1305.5501).

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[talk] Markov Dynamics on Interlacing Arrays

The talk is devoted to the construction of Markov dynamics on interlacing arrays which act nicely on the Macdonald measures. In a particular case one gets $q$-deformed Robinson-Schensted insertion algorithms. The talk is based on [12], [13], see also [14], [15].

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[paper] Nearest neighbor Markov dynamics on Macdonald processes


Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters $(q,t)$, where $0\le q, t < 1$. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes.

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[paper] The Boundary of the Gelfand-Tsetlin Graph: New Proof of Borodin-Olshanski’s Formula, and its q-analogue


In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space.

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[paper] Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field


We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice.

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[talk] Random 3D surfaces and their asymptotic behavior

The talk describes the results of [9], [10], and [11] on asymptotic behavior of random lozenge tilings via determinantal structure and double contour integral formulas for the correlation kernel.

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