Abstract:Links between uniform Aztec diamonds and random matrices are numerous in the literature. In particular, it has been established that, after appropriate rescaling, the probability density function of a certain subclass of dominos converges to the GUE-corners (GUE minor) process in the large size limit. We are interested to see whether this result holds when we modify the probability measure on the space of configurations. In the first part, we look at the case of biased Aztec diamonds, where different weights are associated to vertical and horizontal dominos. In the second part, we examine the case of two-periodic weightings. In both cases, we give exact expressions for the discrete probability distributions of the particles on the levels close to the contact point (turning point), valid for any finite Aztec diamond. In the scaling limit, we prove the convergence to GUE-corners with a rescaling that depends on the weighting. We include a discussion of our results in the light of recent results by Berggren and Bradinoff, who found, in a very close setting, a marked GUE-corners process rather than the classical GUE-corners process.
From: Philippe Ruelle [view email]
[v1]
Wed, 17 Sep 2025 15:16:33 UTC (438 KB)
[v2]
Tue, 1 Sep 2026 14:16:55 UTC (759 KB)
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