Breaking Universality
in Dimer Models
University of Virginia
Joint work with
Alisa Knizel (Barnard / Columbia)
arXiv:2507.22011 (opens in new tab)
and Alexey Bufetov, Panagiotis Zografos (Leipzig)
arXiv:2507.08560 (opens in new tab)
lpetrov.cc/lozenge-draw/
Draw a shape and tile by lozenges
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Limit Shape in 3D
pre-sampled
Theorem [Cohn–Kenyon–Propp 2000]:
As the mesh size → 0, the random stepped surface concentrates around a deterministic limit shape. It develops flat and curved parts.
As the mesh size → 0, the random stepped surface concentrates around a deterministic limit shape. It develops flat and curved parts.
3D surfaces projected to \(x+y+z=0\) are lozenge tilings
with three types of lozenges.
More deformation: q-Racah tilings of the hexagon
Orthogonal polynomials:
Classical: Hermite, Laguerre, Jacobi, Legendre, ...
→ if you ask for explicit coefficients, you can climb up to q-discrete ("exotic") world. The ladder is the (q-)Askey scheme
→ At the very top: q-Racah polynomials
→ if you ask for explicit coefficients, you can climb up to q-discrete ("exotic") world. The ladder is the (q-)Askey scheme
→ At the very top: q-Racah polynomials
Exactly solvable measure on boxed surfaces (hexagon tilings)
discovered by
[Borodin-Gorin-Rains 2009],
connected to the q-Racah polynomials
\[
\mathbb{P}(\text{tiling}) = \frac{1}{Z} \prod_{\substack{\text{horiz.}\\\text{lozenges}}} \mathrm{w}(h),\qquad \mathrm{w}(h) = \chi\, q^{h} + \frac{1}{\chi\, q^{h}}
\]
\(h\) — height, centered around the middle of the hexagon
\(q \in (0,1]\) — volume tilt, \(\chi > 0\) — interpolation parameter
\(\chi \to 0\) gives \(q^{-\text{vol}}\), \(\chi \to +\infty\) gives \(q^{+\text{vol}}\)
\(q \in (0,1]\) — volume tilt, \(\chi > 0\) — interpolation parameter
\(\chi \to 0\) gives \(q^{-\text{vol}}\), \(\chi \to +\infty\) gives \(q^{+\text{vol}}\)
Results [BGR 2009], [Dimitrov-Knizel 2019], [Gorin-Huang 2022], [Duits-Duse-Liu 2023]:
Same local limits (pure states) and GFF fluctuations as \(q = e^{-\gamma/N}\), but different limit shape
Example. Counting horizontal lozenges by centered height:
Particles in two dimensions in a double-well potential
| \(h = -2.5\): | 3 |
| \(h = -1.5\): | 2 |
| \(h = -0.5\): | 4 |
| \(h = \phantom{-}0.5\): | 3 |
| \(h = \phantom{-}1.5\): | 2 |
| \(h = \phantom{-}2.5\): | 2 |
q-Racah Orthogonal Polynomial Ensemble
Vertical slice \(\to\) \(N\) noncolliding paths
Correlation kernel:
\(\displaystyle K_t(x, y) = \sum_{n=0}^{N-1} f_n^t(x) f_n^t(y)\)
\(f_n^t\): orthonormal q-Racah polynomials on slice \(t\)
\(K_t\): spectral projection for q-difference operator
Key observation [Borodin-Gorin-Rains 2009]
The marginal distribution of paths on a vertical slice is the q-Racah orthogonal polynomial ensemble.
This is basically how the deformation was discovered.
Joint distribution:
\(\displaystyle \mathfrak{R}^{qR(N)}_{M,\alpha,\beta,\gamma,\delta}(x_1, \ldots, x_N) = \frac{1}{Z} \prod_{1 \le i < j \le N} (\mu(x_i) - \mu(x_j))^2 \prod_{i=1}^N w^{qR}(x_i)\)
where \(\mu(x) = q^{-x} + \gamma\delta q^{x+1}\)
Weight function:
\(\displaystyle w^{qR}(x) := \frac{(\alpha q; q)_x (\beta\delta q; q)_x (\gamma q; q)_x (\gamma\delta q; q)_x}{(q; q)_x (\alpha^{-1}\gamma\delta q; q)_x (\beta^{-1}\gamma q; q)_x (\delta q; q)_x} \cdot \frac{(1 - \gamma\delta q^{2x+1})}{(\alpha\beta q)^x (1 - \gamma\delta q)}\)
\(\displaystyle (a; q)_k := (1-a)(1-aq) \cdots (1-aq^{k-1})\) is the q-Pochhammer symbol
Parameters:
\((M; \alpha, \beta, \gamma, \delta; q, \chi)\), \(\gamma = q^{-M-1}\)
\(\displaystyle M = t + N - 1,\ \alpha = q^{-S-N},\ \beta = q^{S-T-N},\ \gamma = q^{-t-N},\ \delta = -\chi^2 q^{-S+N}\)
Correlations:
\(\displaystyle \mathbb{P}(\text{particles at } p_1, \ldots, p_m) = \det\bigl[K_t(p_i, p_j)\bigr]_{i,j=1}^{m}\)
Diagonally Layered Disorder
Model:
Sample i.i.d. parameters \(\boldsymbol\beta_1,\ldots,\boldsymbol\beta_M\in(0,1)\). The nontrivial edge weight in diagonal layer \(j\) is
Sample i.i.d. parameters \(\boldsymbol\beta_1,\ldots,\boldsymbol\beta_M\in(0,1)\). The nontrivial edge weight in diagonal layer \(j\) is
\(\displaystyle W_j=\frac{\boldsymbol\beta_j}{1-\boldsymbol\beta_j}\)
Unlabeled edges have weight \(1\). The raw weights \(W_j\) may be heavy-tailed.
Probability measure:
\(\displaystyle \mathbf{P}(D) = \frac{1}{Z} \prod_{e \in D} \operatorname{weight}(e)\)
Two regimes as \(M \to \infty\): [Bufetov–P.–Zografos 2025]
- Critical vanishing variance:
\(M\operatorname{Var}(\boldsymbol\beta_j)\to\sigma^2\), \(\boldsymbol\beta_j\to\beta\).
Same limit shape as deterministic \(W=\beta/(1-\beta)\). Global/moment covariance: GFF + Brownian term. - Fixed law of \(\boldsymbol\beta_j\), independent of \(M\).
New limit shape from \(\mathcal B\). Global moments scale \(\sqrt{M}\), with Brownian-in-level covariance.
Schur Generating Functions for Random Environment
Schur generating function
[Gorin–Panova 2012], [Bufetov–Gorin 2013]:
\(\displaystyle S_{\rho}(u_1,\ldots,u_N) = \sum_{\lambda} \rho(\lambda) \frac{s_\lambda(u_1,\ldots,u_N)}{s_\lambda(1^N)}\)
where \(\rho\) is the distribution of a partition \(\lambda\), and \(u_j \in \mathbb{C}\)
For random environment
[Bufetov–P.–Zografos 2025]:
The annealed SGF has an explicit product form (\(M\) is the size of the Aztec diamond, \(N\) is the level):
The annealed SGF has an explicit product form (\(M\) is the size of the Aztec diamond, \(N\) is the level):
\(\displaystyle
\begin{aligned}
S_{\rho_N}(x_1,\ldots,x_N)
&= \operatorname{\mathbf E}_{\boldsymbol{\mathcal B}_M}
\prod_{j=N+1}^{M}\prod_{i=1}^{N}
(1-\boldsymbol\beta_j+x_i\boldsymbol\beta_j) \\
&=\left(\operatorname{\mathbf E}_{\boldsymbol{\mathcal B}_M}
\prod_{i=1}^{N}(1-\boldsymbol\beta+x_i\boldsymbol\beta)
\right)^{M-N}.
\end{aligned}
\)
where \(\boldsymbol\beta,\boldsymbol\beta_j\sim\boldsymbol{\mathcal B}_M\) are independent, and the raw edge weight is \(W=\boldsymbol\beta/(1-\boldsymbol\beta)\). No moment assumptions on \(W\): it can be heavy-tailed.
Asymptotics of the \(N\)-th root of the SGF as \(N\to\infty\) yield LLN and CLT for
the height function of the domino tilings.
Limit Shapes Under Random Weights
Limit shape equation:
\((\alpha, y) \in [0,1]^2\) are coordinates in the Aztec diamond. Solve for \(z\) in the upper half-plane:
\((\alpha, y) \in [0,1]^2\) are coordinates in the Aztec diamond. Solve for \(z\) in the upper half-plane:
\(\displaystyle \frac{z}{z-1} + (\alpha^{-1}-1) z \, \mathbf{E}_{\mathcal B}\frac{W}{1+W z} = \alpha^{-1} y\)
(recall \(W=\boldsymbol\beta/(1-\boldsymbol\beta)\))
(recall \(W=\boldsymbol\beta/(1-\boldsymbol\beta)\))
Limiting density:
\(\displaystyle (\alpha,y)\mapsto \frac{1}{\pi}\,\mathrm{Arg}\bigl(z(\alpha,y)\bigr)\)
Arctic curve
is the locus of double roots in \(z\), and we take real \(z\) as the parametrization of the frozen boundary.
Bernoulli
\(\tfrac12\) and \(5\)
\(\tfrac12\) and \(5\)
Uniform
\(W_i \sim [0,2]\)
\(W_i \sim [0,2]\)
Straight-Layered Disorder
Straight-layered disorder:
i.i.d. weights along horizontal layers — a 45° rotation from the diagonal case.
\(W_i=0.3\) or \(5\) with prob \(\tfrac{1}{2}\).
[Moulard–Toninelli 2025, arXiv:2507.11964, on torus]: studied singularities of the free energy, showed correlations decay as \(e^{-\sqrt{\text{distance}}}\) instead of polynomially.
Strictly periodic
pre-sampled
Gauge equivalent to 2×2 periodic model
[Chhita–Johansson 2014]
Random layered
pre-sampled
IID Gamma-Distributed Weights
Model:
[Duits–Van Peski 2025, arXiv:2512.03033]
\(\alpha\) edges \(\sim \Gamma(\alpha, 1)\), \(\beta\) edges \(\sim \Gamma(\beta, 1)\). Gamma is the unique distribution preserved by the shuffling algorithm. Turning point fluctuations scale as \(n^{2/3}\) (vs \(n^{1/2}\) for deterministic weights). Connection to integrable polymers (log-Gamma, strict-weak).
\(\alpha\) edges \(\sim \Gamma(\alpha, 1)\), \(\beta\) edges \(\sim \Gamma(\beta, 1)\). Gamma is the unique distribution preserved by the shuffling algorithm. Turning point fluctuations scale as \(n^{2/3}\) (vs \(n^{1/2}\) for deterministic weights). Connection to integrable polymers (log-Gamma, strict-weak).
pre-sampled
\(\alpha = 0.2, \beta = 0.25\)
pre-sampled
\(\alpha = 1.0, \beta = 1.0\)