Local statistics of random domino tilings of L-shaped Aztec domains: unfrozen connecting paths and correlation kernels at the arctic curve and its cusps

Charlier and Claeys (arXiv:2512.20388) proved uniform large-$N$ asymptotics for the weighted number of domino tilings of L-shaped Aztec domains — an Aztec diamond with an approximately rectangular staircase corner removed. When the removed corner is almost maximal (fixed $k$), the count is asymptotically the product of the counts of two disjoint smaller Aztec diamonds and an explicit factor encoding $k$ thin "unfrozen" paths of dominoes that cross the frozen region and connect the two sub-diamonds; when the removed corner is large, the liquid region is bounded by an explicitly known arctic curve with two cusp points. The authors explicitly pose the problem of understanding the local statistics beyond the total count: the asymptotic locations and correlations of the unfrozen paths (Remark 1.2), and the local correlation functions of the tiling, expected to be governed by the Airy process near smooth points of the arctic curve and by a cusp-type process at the cusps (Remark 1.8). This record states those two questions: rigorously derive the limiting local correlation kernels and point processes in the large-corner regime, including edge and cusp limits (Remark 1.8), and characterize the limiting locations and correlation structure of the $k$ unfrozen connecting paths in the almost-maximal regime (Remark 1.2). Both questions remain open in the literature.

Background

A domino is a $1\times 2$ or $2\times 1$ rectangle joining two adjacent unit squares of the square lattice, and a domino tiling is a collection of dominoes covering a given region exactly, with disjoint interiors. The Aztec diamond $A_N$ of order $N$ is the union of the $2N(N+1)$ unit squares of the square lattice that lie entirely inside the region $|x|+|y|\le N+1$; it is the domino-tiling counterpart of the lozenge-tiling hexagon. A uniformly random domino tiling of $A_N$ exhibits the arctic phenomenon: with probability tending to one as $N\to\infty$, the tiling is frozen (all dominoes aligned) in four corner "polar" regions and disordered in a central "rough" (liquid) region, separated by a sharp interface — the arctic curve — which is a circle in the uniform case (the arctic circle theorem of Jockusch, Propp, and Shor) and an ellipse for the natural weighting described next. Elkies, Kuperberg, Larsen, and Propp showed that the weighted partition function $\sum_T a^{v(T)}$, where $v(T)$ is the number of vertical dominoes of the tiling $T$ and $0<a\le1$, equals $(1+a^2)^{N(N+1)/2}$; $a=1$ recovers the uniform measure.

Domino tilings of Aztec-type regions are equivalently ensembles of non-intersecting lattice paths, and under this correspondence the tiling is a determinantal point process: a random point set whose joint intensities are given by determinants of a single correlation kernel (Johansson, 2002). The asymptotic analysis of such kernels near the arctic boundary is the gateway to local universality: in the full Aztec diamond, the rescaled edge of the rough region converges to the Airy process, the determinantal process with the extended Airy kernel that also governs edge eigenvalues of large random matrices (Johansson, 2005).

This record concerns L-shaped Aztec domains $A_N^{m,k}$ (Charlier and Claeys, 2025): the region obtained from $A_N$ by removing, at a corner, a staircase "approximately rectangular" region with $N+1-m$ unit steps along one side and $m-k+1$ along the other, where $1\le m\le N$ and $1\le k\le m+1$. The source treats this domain together with a companion variant $\tilde{A}_N^{m,k}$, obtained from $A_N^{m+1,k+1}$ by deleting a single boundary unit segment, which realizes the complementary staircase parity; a binary parameter $\epsilon\in\{0,1\}$ selects between the two domains. Domino tilings of $A_N^{m,k}$ correspond to the free-fermion six-vertex model (a classical spin model on the square lattice whose determinantal "free-fermion" case is equivalent to domino tilings) in an L-shaped domain. Colomo and Pronko computed the leading-order free energy of that model and identified a third-order phase transition in the size of the removed region; Kitaev and Pronko later obtained the full asymptotic expansion of the tiling count in a special symmetric case; and Colomo, Pronko, and Sportiello computed the arctic curve of the L-shaped geometry explicitly by the tangent method (a technique for determining frozen boundaries from the endpoints of long non-intersecting paths). Unlike the ellipse of the full diamond, this arctic curve has two cusp points — non-smooth points where the frozen boundary turns. Charlier and Claeys (arXiv:2502.19096, published in the Journal of the London Mathematical Society, 2026) developed a Riemann–Hilbert approach — a steepest-descent technique from integrable-systems and random-matrix theory for extracting asymptotics from analytic factorizations of determinants — to gap probabilities (probabilities that a region is entirely frozen) of these tiling ensembles, and in arXiv:2512.20388 they proved uniform asymptotics for the weighted count $F_N^{m,k}(a;\epsilon)$ (the sum of $a^{v(T)}$ over tilings $T$ of the selected domain) in all regimes of the removed corner: small corner (count close to that of the full diamond), critical corner (a phase transition described by the Tracy–Widom distribution, the universal edge distribution of random matrix theory), large corner ($k/N\in(0,\kappa_2(\mu))$ with $\kappa_2$ explicit, for $\mu=m/N$ above $\frac{a^{2}}{1+a^{2}}$), and almost-maximal corner (fixed $k$).

In the almost-maximal regime their Theorem 1.1 states, for fixed $k\ge0$ and $\mu=m/N$ away from $0$ and $1$, that

$$F_N^{m,k+1}(a;\epsilon)=F_{m-\epsilon}(a;\epsilon)F_{N-m}(a;\epsilon)\,e^{k\varphi_0(x_0)N}\,N^{-k^2/2}\times(\text{explicit constant})\times(1+O(N^{-1/2})),$$

where $F_n(a;\epsilon)=(1+a^{2})^{n(n+1)/2}$ is the weighted partition function of the full Aztec diamond of order $n$, and $\varphi_0$ and $x_0$ are explicit. For $k=0$ the domain splits into two exactly disjoint Aztec diamonds, and each additional unit of $k$ multiplies the number of tilings by roughly $e^{\varphi_0(x_0)N}$. Simulations indicate that this growth corresponds to $k$ thin "unfrozen paths" of non-frozen dominoes crossing the frozen sea and connecting the two sub-diamonds — a phenomenon the authors describe as the first observation and quantification of unfreezing by a small number of paths over a long distance, with distant relatives in the tacnode process of overlapping Aztec diamonds (Adler, Johansson, and van Moerbeke), where two clouds of non-intersecting paths touch at a single point.

For the full Aztec diamond, the boundary limit is the Airy process (Johansson, 2005), and for lozenge tilings of polygons the statistics are likewise understood in depth: the global height fluctuations converge to the Gaussian free field — the massless Gaussian field of statistical mechanics — via an explicit double-contour representation of the correlation kernel (Petrov, 2015); at cusps of polygonal lozenge tilings, Pearcey universality holds (Huang, Yang, and Zhang, 2024), the Pearcey process being the determinantal process describing generic caustic cusps; and the cusp-Airy process, a two-sided extension of the Airy point process, was derived by Duse, Johansson, and Metcalfe for a class of lozenge-tiling models with cusps. Very recently, Adler and van Moerbeke (2026) showed that at the split cusps of lozenge tilings of hexagons with cuts, yet another process, the transversal cusp-Airy, appears — so the universality class at cusp points is subtle and model-dependent. For the L-shaped Aztec domains, by contrast, nothing beyond the total count is known: as Charlier and Claeys write, "Finer statistics like local correlation functions have not been analyzed so far" (Remark 1.8), and "It would be interesting to understand not only the number of tilings, but also the locations and correlations of the unfrozen paths" (Remark 1.2).

Problem Statement

The problem, posed by Charlier and Claeys in Remarks 1.2 and 1.8 of arXiv:2512.20388, is to understand the local statistics of random domino tilings of L-shaped Aztec domains beyond the total count, for the measure assigning each tiling $T$ of $A_N^{m,k}$ (or of the variant $\tilde{A}_N^{m,k}$) the weight $a^{v(T)}$ with $0<a\le1$ and $v(T)$ the number of vertical dominoes. The source poses two open questions, one for each regime (Remark 1.2 on the unfrozen paths, Remark 1.8 on the local correlation functions); they are stated together here because they concern the same tiling ensemble.

(1) Unfrozen connecting paths (almost-maximal removed corner). Fix an integer $k\ge1$ and $\mu=m/N\in(0,1)$ bounded away from $0$ and $1$ (the range in which Theorem 1.1 of the source is uniform), and let $N\to\infty$. Here $k$ is counted as in Theorem 1.1 and Remark 1.2 of the source: the domain is $A_N^{m,k+1}$, the case $k=0$ is the domain $A_N^{m,1}$, which splits into two exactly disjoint Aztec diamonds, and each $k\ge1$ corresponds to $k$ unfrozen connecting paths. Give a rigorous formulation of the $k$ "unfrozen paths" connecting the two smaller Aztec diamonds (in the source these are observed in simulations as apparent thin corridors of non-frozen dominoes, so a definition via the non-intersecting-path representation or the height function (the integer-valued function on lattice vertices that encodes a tiling and changes by fixed increments across each domino) is a legitimate part of the answer), and then characterize their asymptotic statistics: the macroscopic routes and transverse fluctuations of the paths (their typical locations) and their correlation structure — for instance, but not necessarily, by identifying the limiting point process or line ensemble (a random ordered stack of curves, as for the Airy line ensemble) obtained from the paths in suitable scaling coordinates.

(2) Local correlation functions (large removed corner). Let $\mu=m/N\in\big(\frac{a^{2}}{1+a^{2}},1\big)$ — the range in which the source places the transition where the removed region crosses the arctic curve; for smaller $\mu$ the removed corner immediately crosses the rough region and this regime does not occur — and let $\kappa=k/N\in(0,\kappa_2(\mu))$, where $\kappa_2(\mu)=\frac{a^{2}(2\mu-1)+2a\sqrt{\mu(1-\mu)}}{1+a^{2}}$ is the explicit threshold of Charlier and Claeys at which the removed region crosses the arctic curve (at $\mu=\frac{a^{2}}{1+a^{2}}$ one has $\kappa_2=\mu$, and $\kappa_2(\mu)\in(0,\mu)$ on this range), and below which the removed corner still overlaps the rough region. In this item, $k$ is the domain index of $A_N^{m,k}$ (Theorem 1.4 of the source uses $\kappa_+=(2k+1)/(2N)=\kappa+O(N^{-1})$). Derive the limiting local correlation kernel of the determinantal tiling ensemble: for points in the liquid region; near smooth points of the arctic curve (computed explicitly by Colomo, Pronko, and Sportiello), where Airy-process behavior is expected as in the full Aztec diamond; and near the two cusp points of the arctic curve, where the expected behavior is "related to the cusp-Airy process" of Duse, Johansson, and Metcalfe, but the correct limiting process — cusp-Airy, Pearcey, transversal cusp-Airy, or another — must be identified by the analysis itself. The scaling coordinates and limiting kernels should be explicit.

The source poses these as two open questions, Remark 1.2 for regime (1) and Remark 1.8 for regime (2); each asks for rigorous theorems establishing the stated limiting objects. In regime (2), identifying the correct edge process at the cusps is part of the answer, not a hypothesis.

Questions (1) and (2) are independent of one another: a rigorous resolution of either one constitutes a valid answer to this record, while a complete resolution requires both. Question (1) is resolved once the macroscopic limiting trajectories of the $k$ unfrozen paths and their joint limiting fluctuation/correlation law are established (a limiting point process or line ensemble is one acceptable form); question (2) is resolved once each of the three limits — the liquid region, smooth points of the arctic curve, and the two cusp points — is established.

References

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