# Lozenge Tilings and Hurwitz Numbers

@article{Novak2014LozengeTA, title={Lozenge Tilings and Hurwitz Numbers}, author={Jonathan Novak}, journal={Journal of Statistical Physics}, year={2014}, volume={161}, pages={509-517} }

We give a new proof of the fact that, near a turning point of the frozen boundary, the vertical tiles in a uniformly random lozenge tiling of a large sawtooth domain are distributed like the eigenvalues of a GUE random matrix. Our argument uses none of the standard tools of integrable probability. In their place, it uses a combinatorial interpretation of the Harish-Chandra/Itzykson-Zuber integral as a generating function for desymmetrized Hurwitz numbers.

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#### References

SHOWING 1-10 OF 30 REFERENCES

New scaling of Itzykson-Zuber integrals

- Mathematics
- 2007

Abstract We study asymptotics of the Itzykson–Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in… Expand

Correlations for the Novak process

- Mathematics
- 2012

We study random lozenge tilings of a certain shape in the plane called the Novak half-hexagon, and compute the correlation functions for this process. This model was introduced by Nordenstam and… Expand

Towards the geometry of double Hurwitz numbers

- Mathematics
- 2003

Abstract Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞ , and the branching over 0 and ∞ specified by… Expand

Toda equations for Hurwitz numbers

- Mathematics, Physics
- 2000

We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings… Expand

Monotone Hurwitz Numbers and the HCIZ Integral

- Mathematics, Physics
- 2011

In this article, we study the topological expansion of the Harish-Chandra-Itzykson-Zuber matrix model. We prove three types of results concerning the free energy of the HCIZ model. First, at the… Expand

Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

- Mathematics, Physics
- 2015

We develop a new method for studying the asymptotics of symmetric polynomials of representation- theoretic origin as the number of variables tends to infinity. Several applications of our method are… Expand

The spectrum of coupled random matrices

- Physics, Mathematics
- 1999

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this… Expand

The Spectrum of coupled random matrices

- Mathematics
- 1999

The study of the spectrum of coupled random matrices has received rather little attention. To the best of our knowledge, coupled random matrices have been studied, to some extent, by Mehta. In this… Expand

Parking Functions of Types A and B

- Computer Science, Mathematics
- Electron. J. Comb.
- 2002

The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use… Expand

Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

- Mathematics, Physics
- 2012

A Gelfand–Tsetlin scheme of depth $$N$$N is a triangular array with $$m$$m integers at level $$m$$m, $$m=1,\ldots ,N$$m=1,…,N, subject to certain interlacing constraints. We study the ensemble of… Expand