Biased Aztec Diamonds: Weighted Domino Tiling

• Biased Aztec Diamonds are weighted variants of the classic domino tiling model that incorporate diverse bias mechanisms, including orientation bias, periodic weights, spatial inhomogeneity, and defects.
• The analysis employs advanced methods such as Kasteleyn theory, determinantal point processes, and saddle-point asymptotics to derive explicit formulas for partition functions, correlation kernels, and limit shapes.
• Introducing bias leads to new combinatorial and geometric phenomena, such as modified phase diagrams, arctic boundaries, and random matrix asymptotics, which extend classical results in tiling theory.

Biased Aztec diamonds are weighted or constrained variants of the Aztec diamond domino-tiling model in which the uniform measure is replaced by a measure that distinguishes orientations, locations, periodic classes, interfaces, defects, or random environments. In the published literature, bias appears in several non-equivalent but closely related forms: horizontal and vertical dominoes can carry different weights; the weights can be periodic or two-periodic; the domain can be split into regions with different weightings; admissible tile sets can be restricted; or boundary squares can be removed. These models remain closely tied to dimer and perfect-matching formalisms, and much of their analysis proceeds through Kasteleyn theory, determinantal point processes, interlacing arrays, and saddle-point asymptotics (Boutillier et al., 2024).

1. Weighted formulations of bias

A basic orientation-biased model assigns weight $1$ to each horizontal domino and weight $\sqrt{\lambda}$ to each vertical domino, so that the weight of a tiling configuration $c$ is

$w(c)=\lambda^{N_v(c)/2},$

where $N_v(c)$ is the total number of vertical dominos in $c$, and the probability of $c$ is $w(c)/Z_n$. When $\lambda=1$, one recovers the uniform case (Robert et al., 17 Sep 2025). A more structured version is the biased $2\times 2$ periodic Aztec diamond, with parameters $\sqrt{\lambda}$0 and $\sqrt{\lambda}$1, where $\sqrt{\lambda}$2 suppresses vertical dominos, favoring horizontal ones, while $\sqrt{\lambda}$3 introduces $\sqrt{\lambda}$4 periodicity (Borodin et al., 2022).

A second major class consists of spatially inhomogeneous models. In the split two-periodic Aztec diamond, the left and right halves carry different two-periodic parameters $\sqrt{\lambda}$5 and $\sqrt{\lambda}$6, thereby breaking periodicity in one direction along a vertical interface (Shea, 25 Feb 2025). In the $\sqrt{\lambda}$7-split two-periodic Aztec diamond, the domain is divided into two unequal regions of sizes proportional to $\sqrt{\lambda}$8 and $\sqrt{\lambda}$9, with the interface at $c$0, and the larger and smaller sides are termed the dominant and non-dominant sides (Shea, 17 Jun 2026).

Bias also arises through changes in admissible configurations rather than edge weights. Restricting the allowed tile set can be interpreted as introducing biases into the tiling model, as in Aztec diamonds tiled by dominos together with square or skew tetrominos (Propp, 2022). Boundary defects furnish another form of bias: the dented region $c$1 is obtained from the Aztec diamond by removing the $c$2-th square from the southwestern side and the $c$3-th from the southeastern side (Ciucu et al., 2023). Finally, randomness itself can be the source of bias: in the Gamma-disordered model, edge weights are random variables $c$4 and $c$5 (Duits et al., 2 Dec 2025).

Bias mechanism	Representative parameters	Reported effect
Orientation bias	$c$6 for horizontal, $c$7 for vertical	Elliptic arctic geometry and GUE-corners scaling
$c$8 periodic bias	$c$9	Frozen, rough, and smooth disordered regions
Split or $w(c)=\lambda^{N_v(c)/2},$0-split weights	$w(c)=\lambda^{N_v(c)/2},$1	Interface-dependent kernels and asymmetric limit shapes
Boundary dents or defects	$w(c)=\lambda^{N_v(c)/2},$2 or defect position $w(c)=\lambda^{N_v(c)/2},$3	Delannoy asymptotics and symmetry phenomena
Random disorder	Gamma parameters $w(c)=\lambda^{N_v(c)/2},$4	No free-energy phase transition and polymer correspondences

2. Exact solvability, Kasteleyn theory, and correlation kernels

The modern theory of biased Aztec diamonds is formulated largely through dimers on the Aztec diamond graph. In this setting, a dimer configuration is a perfect matching, and correlations are encoded by the inverse Kasteleyn matrix. A particularly general result is that any dimer model on the Aztec diamond with arbitrary positive edge weights is gauge equivalent to a model with Fock's weights. In that framework, if an edge $w(c)=\lambda^{N_v(c)/2},$5 is crossed by train-tracks with angles $w(c)=\lambda^{N_v(c)/2},$6, and $w(c)=\lambda^{N_v(c)/2},$7 are the adjacent dual faces, then the Kasteleyn entry is

$w(c)=\lambda^{N_v(c)/2},$8

The same work provides an explicit formula for the inverse Kasteleyn matrix and a product recurrence for the partition function; in the uniform genus $w(c)=\lambda^{N_v(c)/2},$9 case, the partition function is $N_v(c)$0, and Stanley's product formula is recovered as a special case (Boutillier et al., 2024).

For the biased $N_v(c)$1 periodic model, the determinantal kernel is expressed as a double contour integral whose Wiener–Hopf factorization is governed by a linear flow on the elliptic curve

$N_v(c)$2

For special choices of parameters, this flow is periodic; in the period-six example, the rough-disordered boundary is shown to be an algebraic curve of degree eight (Borodin et al., 2022).

In split models, the kernel becomes piecewise because the asymptotics depend on the positions of points relative to the interface. The split two-periodic Aztec diamond kernel is obtained by extending methods based on the Eynard–Mehta theorem and a Wiener–Hopf factorization, while the $N_v(c)$3-split model yields an explicit $N_v(c)$4 kernel whose form differs on the two sides of the interface and generalizes the kernels for the uniform, two-periodic, and split two-periodic models (Shea, 25 Feb 2025). In both cases, the kernel is the basic object from which local probabilities, asymptotic phases, and interface effects are derived (Shea, 17 Jun 2026).

3. Limit shapes, frozen boundaries, and phase structure

A central asymptotic phenomenon is concentration of the random height function around a deterministic limit shape. For rectangular Aztec diamonds, the normalized height function $N_v(c)$5 converges in probability to a deterministic function $N_v(c)$6, and the frozen boundary separating liquid and frozen regions is algebraic and rationally parameterized. For special classes of examples, the boundary can be written in terms of

$N_v(c)$7

with $N_v(c)$8 encoding the boundary data. The same source states that the approach extends to weighted tilings: for biased models and piecewise constant boundary data, the frozen boundary remains algebraic and may be a union of finitely many algebraic curves (Bufetov et al., 2016).

Two-periodic weights introduce a richer phase diagram than the uniform case. When $N_v(c)$9 is fixed in the two-periodic weighted dimer model, the large system has frozen regions near the corners, a flat ordered central region, and a disordered region separating the flat and frozen phases. The original arctic circle is replaced by algebraic curves separating these three phases (Bain, 2022). In the biased $c$0 periodic model, the large-scale decomposition is described as frozen, rough disordered, and smooth disordered, and the boundary between rough and smooth disordered regions is algebraic; in the period-six example it has degree eight (Borodin et al., 2022).

Spatially split models break the left-right symmetry of the phase diagram. In the $c$1-split two-periodic Aztec diamond, the dominant side has a limit shape identical to that of the standard two-periodic model with parameter $c$2, while the non-dominant side has a novel limit shape depending on $c$3, $c$4, and $c$5. Conjecture 1 states that the non-dominant side is controlled by the saddle function

$c$6

and Conjecture 2 states that a smooth region appears on the left if and only if $c$7 (Shea, 17 Jun 2026).

The general weighted theory connects limit shapes to spectral geometry. For periodic Fock's weights, the liquid, gas, and frozen regions correspond to components of the spectral curve, and the correspondence between the limit shape and the amoeba of the spectral curve extends to non-generic weights, including singular or degenerate cases (Boutillier et al., 2024).

4. Fluctuations, mesoscopic limits, and random-matrix asymptotics

Beyond the law of large numbers, several fluctuation regimes are known. For rectangular Aztec diamonds, centered and scaled height fluctuations converge to the Gaussian Free Field on the liquid region with Dirichlet boundary conditions (Bufetov et al., 2016). This is one of the main universal fluctuation statements available for weighted Aztec-type domains.

Two-periodic weighting produces additional mesoscopic structure. In the scaling regime

$c$8

the width of the flat region is of the same order as the correlation length. In this limit there is no macroscopic flat region, the leading local dimer density is

$c$9

and more precisely

$c$0

The sub-leading behavior is described by a new double integral process rather than the Airy process (Bain, 2022).

In the orientation-biased model with horizontal weight $c$1 and vertical weight $c$2, the induced interlaced particle process has explicit marginals. On a single level,

$c$3

which is the Krawtchouk ensemble with $c$4. After centering and scaling by

$c$5

the point process on finitely many levels converges to the GUE-corners process (Robert et al., 17 Sep 2025).

Random disorder changes the fluctuation theory qualitatively. In the Gamma-disordered Aztec diamond, there is no phase transition at the level of the free energy, and the free-energy fluctuations satisfy a central limit theorem. At the boundary, the west, north, and south turning points have fluctuations of order $c$6, in contrast to the $c$7 fluctuations for deterministic weights, while the east turning point retains $c$8 Gaussian fluctuations. The same work identifies exact distributional equalities between dimer marginals and path locations in stationary log-Gamma, strict-weak, Beta, and hybrid polymers (Duits et al., 2 Dec 2025).

5. Interfaces, defects, and deformed domains

The split two-periodic Aztec diamond isolates the effect of an abrupt change in weighting. Its local asymptotics agree with the typical two-periodic model in the highest order, but the sub-leading order terms are affected. Away from the interface, the leading local behavior is governed by the weighting of the region under consideration; near the interface and in strong-coupling regions, slower $c$9 corrections may persist (Shea, 25 Feb 2025). The $w(c)/Z_n$0-split model sharpens this asymmetry: the dominant side is insensitive to the weighting on the non-dominant side, while the non-dominant side reflects both weightings and the interface position. As $w(c)/Z_n$1, the dominant side becomes frozen-smooth, and on the non-dominant side the kernel simplifies to half of a rescaled two-periodic Aztec diamond (Shea, 17 Jun 2026).

Boundary dents provide an explicitly enumerable defect model. For the region $w(c)/Z_n$2,

$w(c)/Z_n$3

and for fixed $w(c)/Z_n$4,

$w(c)/Z_n$5

where $w(c)/Z_n$6 is a Delannoy number. If the line segment connecting the scaled defect locations lies outside the inscribed circle, then

$w(c)/Z_n$7

whereas if the segment crosses the circle, the limit is $w(c)/Z_n$8. Under the assumption that an arctic curve exists, this yields a derivation of the arctic circle for Aztec diamonds (Ciucu et al., 2023).

Odd-order off-diagonally symmetric tilings with a single boundary defect furnish a different defect class. If $w(c)/Z_n$9 denotes the set of such tilings with the unit square at position $\lambda=1$0 removed, then

$\lambda=1$1

More generally, the relevant enumerations admit Pfaffian formulas, and the associated matrix equations have coefficients given by Delannoy numbers (Lee, 2024).

Domain deformation leads to further phase transitions. For an L-shaped subset $\lambda=1$2 with weighted generating function

$\lambda=1$3

small removed corners leave the generating function close to that of the full diamond; when the cut reaches the arctic ellipse at

$\lambda=1$4

a Tracy–Widom regime occurs; deeper cuts produce rough-phase asymptotics with explicit $\lambda=1$5, $\lambda=1$6, and $\lambda=1$7 terms; and almost maximal cuts are asymptotic to two smaller disjoint Aztec diamonds (Charlier et al., 23 Dec 2025).

6. Enumerative, combinatorial, and geometric extensions

Bias can also be introduced through multi-layer or non-domino combinatorics. In $\lambda=1$8-tilings of the Aztec diamond, one studies $\lambda=1$9-tuples of domino tilings with interaction weight $2\times 2$0 for coupled dominos. The generating polynomial is

$2\times 2$1

At $2\times 2$2, there is a weight-preserving bijection between $2\times 2$3-tilings with no coupled dominos and ordinary $2\times 2$4-tilings, and explicit arctic curves are computed for $2\times 2$5 and $2\times 2$6 (Corteel et al., 2022).

Restriction of the tile set yields striking arithmetic effects. For the 63 possible non-empty subsets of the six basic tiles, roughly one third yield even numbers of tilings for all $2\times 2$7, while the case of dominos and square tetrominos is the only one found to be always odd. In that model, the conjectures

$2\times 2$8

and

$2\times 2$9

express a proposed $\sqrt{\lambda}$00-adic periodicity and anti-symmetry. For horizontal skew tetrominos and square tetrominos, the conjectured valuation is

$\sqrt{\lambda}$01

These results show that tile-set bias can alter the arithmetic of tiling counts as radically as it alters local geometry (Propp, 2022).

A geometric synthesis of periodic weighting is provided by perfect $\sqrt{\lambda}$02-embeddings of doubly periodic Aztec diamonds. The associated $\sqrt{\lambda}$03-surfaces converge to space-like maximal surfaces in $\sqrt{\lambda}$04. All frozen regions collapse to four boundary points, while each gas region collapses to a distinct light-like cusp in the interior. In the absence of gas regions, the limiting surface lies entirely within $\sqrt{\lambda}$05, but in the general case this is no longer true. The positions of cusps and boundary vertices depend on the precise distribution of edge weights, whereas the global conformal structure coincides with the Kenyon–Okounkov conformal structure. The paper further conjectures that cusp locations encode the shift in the discrete Gaussian component of global fluctuations (Berggren et al., 6 Aug 2025).