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Interlaced Particle Systems and Tilings of the Aztec Diamond

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Abstract

Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions. These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties of the tiling problem, for example that the number of distinct configurations is 2N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system, now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec diamond tilings.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria, 3010, Australia

    Benjamin J. Fleming & Peter J. Forrester

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  1. Benjamin J. Fleming
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  2. Peter J. Forrester
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Correspondence to Peter J. Forrester.

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Fleming, B.J., Forrester, P.J. Interlaced Particle Systems and Tilings of the Aztec Diamond. J Stat Phys 142, 441–459 (2011). https://doi.org/10.1007/s10955-011-0121-2

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  • DOI: https://doi.org/10.1007/s10955-011-0121-2

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