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Kempner Colloquium
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\begin{Huge}\textsc{Random lozenge tilings and other \\ \vskip .1 in integrable probabilistic models}
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\begin{Large}\textbf{Leonid Petrov}
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(Northeastern)
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I will discuss the�probabilistic model of randomly tiling a hexagon drawn on
the regular triangular lattice�by lozenges of three types (equivalent
formulations: dimer models on the honeycomb lattice, or random 3D stepped
surfaces glued out of 1x1x1 boxes).�This model has received a significant
attention over the past 20 years (first results - the computation of the
partition function -�date back to P. MacMahon, 100+ years ago).�Kenyon,
Okounkov, and their co-authors (1998-2007) proved the law of large numbers:
when the polygon�is fixed and the mesh of the lattice goes to zero, the random
3D surface concentrates around a deterministic limit shape, which is algebraic.
I will discuss finer asymptotics:�local geometry, behavior of interfaces
between phases (which manifests the Kardar-Parisi-Zhang universality), and
global fluctuations of random surfaces (described by the Gaussian Free Field),
as well as dynamical models associated with random tilings.

I also plan to briefly survey the general phenomenon of ``integrable"
probabilistic models, in which the presence of�explicit formulas describing
their distributions allow for an analysis by�essentially algebraic methods.




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December 17, 2013

4:00 p.m.

MATH 350

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