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}
}
  • Jonathan Novak
  • Published 28 July 2014
  • Mathematics, Physics
  • Journal of Statistical Physics
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. 

Figures from this paper

Gaussian Unitary Ensemble in random lozenge tilings
This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever a boundary of the domain has three adjacent straightExpand
Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class
This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in theExpand
Bulk Universality for Random Lozenge Tilings Near Straight Boundaries and for Tensor Products
We prove that the asymptotic of the bulk local statistics in models of random lozenge tilings is universal in the vicinity of straight boundaries of the tiled domains. The result applies to uniformlyExpand
Fluctuations of particle systems determined by Schur generating functions
Abstract We develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems. We introduce and study the notion of the Schur generating function of a randomExpand
DERIVATIVE ASYMPTOTICS OF UNIFORM GELFAND-TSETLIN PATTERNS
Bufetov and Gorin introduced the idea of applying differential operators which are diagonalized by the Schur functions to Schur generating functions, a generalization of probability generatingExpand
On the Complex Asymptotics of the HCIZ and BGW Integrals
In this paper, we prove a longstanding conjecture on the asymptotic behavior of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber (HCIZ) integral, and the Brezin-Gross-WittenExpand
Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to newExpand
On the Convergence of Monotone Hurwitz Generating Functions
Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of severalExpand
GUE corners limit of q-distributed lozenge tilings
We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider eachExpand
Limit shape and height fluctuations of random perfect matchings on square-hexagon lattices
We study asymptotics of perfect matchings on a large class of graphs called the contracting square-hexagon lattice, which is constructed row by row from either a row of a square grid or a row of aExpand
...
1
2
...

References

SHOWING 1-10 OF 30 REFERENCES
New scaling of Itzykson-Zuber integrals
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 inExpand
Correlations for the Novak process
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 andExpand
Towards the geometry of double Hurwitz numbers
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 byExpand
Toda equations for Hurwitz numbers
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 coveringsExpand
Monotone Hurwitz Numbers and the HCIZ Integral
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 theExpand
Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory
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 areExpand
The spectrum of coupled random matrices
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 thisExpand
The Spectrum of coupled random matrices
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 thisExpand
Parking Functions of Types A and B
  • P. Biane
  • 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 useExpand
Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes
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 ofExpand
...
1
2
3
...