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.

[2013/10/30]

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.

[2013/08/15]

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.

[2013/08/14]

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).

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].

[2013/05/23]

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.

[2012/08/16]

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.

[2012/06/22]

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

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.

[2012/02/17]

A Gelfand-Tsetlin scheme of depth $N$ is a triangular array with m integers at level $m$, $m=1,\ldots,N$, subject to certain interlacing constraints. We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed $N$-th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation).