For Aztec diamond tilings with i.i.d. one-periodic edge weights, height function fluctuations are, in the critical regime, GFF plus independent Brownian motion, and in the fixed-variance regime, Brownian motion alone at scale sqrt M.
Asymptotics of Harish-Chandra transform and infinitesimal freeness
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abstract
In the last ten years a technique of Schur generating functions and Harish-Chandra transforms was developed for the study of the asymptotic behavior of discrete particle systems and random matrices. In the current paper we extend this toolbox in several directions. We establish general results which allow to access not only the Law of Large Numbers, but also next terms of the asymptotic expansion of averaged empirical measures. In particular, this allows to obtain an analog of a discrete Baik-Ben Arous-Peche phase transition. A connection with infinitesimal free probability is shown and a quantized version of infinitesimal free probability is introduced. Also, we establish the Law of Large Numbers for several new regimes of growth of a Harish-Chandra transform.
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Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions
For Aztec diamond tilings with i.i.d. one-periodic edge weights, height function fluctuations are, in the critical regime, GFF plus independent Brownian motion, and in the fixed-variance regime, Brownian motion alone at scale sqrt M.