Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains

TL;DR

This paper studies the boundary fluctuations of domino tilings in a restricted Aztec diamond, showing convergence to the Airy$_2$ process conditioned by a parabola, revealing new connections to the hard-edge tacnode process.

Contribution

It introduces a new limit shape analysis for restricted Aztec diamonds and establishes the convergence to the hard-edge tacnode process with explicit statistics.

Findings

Boundary fluctuations converge to the Airy$_2$ process conditioned below a parabola.

The limit process is identified as the hard-edge tacnode process.

Explicit kernels for the finite-dimensional distributions are derived.

Abstract

We consider uniform random domino tilings of the restricted Aztec diamond which is obtained by cutting off an upper triangular part of the Aztec diamond by a horizontal line. The restriction line asymptotically touches the arctic circle that is the limit shape of the north polar region in the unrestricted model. We prove that the rescaled boundary of the north polar region in the restricted domain converges to the Airy$_2$ process conditioned to stay below a parabola with explicit continuous statistics and the finite dimensional distribution kernels. The limit is the hard-edge tacnode process which was first discovered in the framework of non-intersecting Brownian bridges. The proof relies on a random walk representation of the correlation kernel of the non-intersecting line ensemble which corresponds to a random tiling.