Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quenched and Annealed CLTs for the one-periodic Aztec diamond in random environment

Panagiotis Zografos

TL;DR

This work analyzes the asymptotics of random domino tilings of the one-periodic Aztec diamond in random environment, proving a quenched CLT for the height function with Gaussian Free Field fluctuations and deterministic limit shapes. It extends annealed CLTs to non-i.i.d. edge weights, revealing two regimes: a classical $\sqrt{M}$-scale Gaussian field regime and a $M$-scale regime where the fluctuations comprise the GFF plus an independent Gaussian field; both regimes admit explicit covariance formulas. The authors leverage Schur processes with random parameters and differential-operator techniques to derive LLN and CLT results, including multi-level extensions and concrete examples such as i.i.d. weights, Markov chains, and GUE eigenvalues. The results demonstrate that quenched fluctuations preserve the Gaussian Free Field universality while annealed fluctuations encode environmental randomness, providing new limit shapes determined by random edge weights.

Abstract

We study the asymptotic behavior of random dimer coverings of the one-periodic Aztec diamond in random environment. We investigate quenched limit theorems for the height function and we extend annealed limit theorems that were recently studied in [arXiv:2507.08560]. We consider more general choices of random edge weights (independence is not assumed) and we distinguish two cases where the random edge weights satisfy the Central Limit Theorem (CLT) under different scalings. For both cases, we prove convergence to the Gaussian Free Field for the quenched fluctuations. For the annealed version, it had been shown in [arXiv:2507.08560], that Gaussian Free Field fluctuations can be dominated by the much larger fluctuations of the random environment. To access quenched fluctuations we analyze the Schur process with random parameters in a way that allows to prove the annealed CLT for the height function for non i.i.d. weights. We consider specific examples where we determine the asymptotic fluctuations.

Quenched and Annealed CLTs for the one-periodic Aztec diamond in random environment

TL;DR

This work analyzes the asymptotics of random domino tilings of the one-periodic Aztec diamond in random environment, proving a quenched CLT for the height function with Gaussian Free Field fluctuations and deterministic limit shapes. It extends annealed CLTs to non-i.i.d. edge weights, revealing two regimes: a classical -scale Gaussian field regime and a -scale regime where the fluctuations comprise the GFF plus an independent Gaussian field; both regimes admit explicit covariance formulas. The authors leverage Schur processes with random parameters and differential-operator techniques to derive LLN and CLT results, including multi-level extensions and concrete examples such as i.i.d. weights, Markov chains, and GUE eigenvalues. The results demonstrate that quenched fluctuations preserve the Gaussian Free Field universality while annealed fluctuations encode environmental randomness, providing new limit shapes determined by random edge weights.

Abstract

We study the asymptotic behavior of random dimer coverings of the one-periodic Aztec diamond in random environment. We investigate quenched limit theorems for the height function and we extend annealed limit theorems that were recently studied in [arXiv:2507.08560]. We consider more general choices of random edge weights (independence is not assumed) and we distinguish two cases where the random edge weights satisfy the Central Limit Theorem (CLT) under different scalings. For both cases, we prove convergence to the Gaussian Free Field for the quenched fluctuations. For the annealed version, it had been shown in [arXiv:2507.08560], that Gaussian Free Field fluctuations can be dominated by the much larger fluctuations of the random environment. To access quenched fluctuations we analyze the Schur process with random parameters in a way that allows to prove the annealed CLT for the height function for non i.i.d. weights. We consider specific examples where we determine the asymptotic fluctuations.

Paper Structure

This paper contains 20 sections, 20 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. The model
  4. Main results
  5. Preliminaries
  6. Dimer coverings of the Aztec diamond and signatures
  7. The one-periodic Aztec diamond and Schur functions
  8. The Law of Large Numbers
  9. Differential operators
  10. Some technical lemmas
  11. Random weights and limit shape
  12. The Annealed Central Limit Theorem for one level
  13. First limit regime: classical scaling
  14. Computation of the covariance for the first limit regime
  15. Asymptotic normality
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that the random edge weights of the one-periodic Aztec diamond are CLT appropriate. Then, almost surely with respect to the distribution of the random weights, the height function has a limit shape that corresponds to deterministic edge weights. Furthermore, almost surely with respect to the

Figures (1)

  • Figure 1: Left: The Aztec diamond graph of size 3 with one-periodic weights. Right: A dimer covering of the graph. The product of the edge weights for the specific covering is equal to $u_3 v_3 u_2 u_1 v_1^2 x_2^2 x_1 w_3^2 w_2$.

Theorems & Definitions (29)

  • Theorem 1: Informal version of the quenched CLT; Theorem \ref{['NMNm']}
  • Theorem 2: Informal versions of annealed CLTs; Theorems \ref{['nzgamskl']}, \ref{['alloena']}
  • Corollary 1: Brownian motion type fluctuations; Examples \ref{['ex21']}, \ref{['ex22']}
  • Corollary 2: GUE random environment; Example \ref{['ex24']}
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 19 more