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Matrix valued orthogonal polynomials arising from hexagon tilings with 3x3-periodic weightings

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Pith's one-line read For a symmetric 3x3-periodic hexagon tiling, the zeros of the associated matrix-valued orthogonal polynomials converge to the unit circle with an explicitly identified density.

desk verdict First strong asymptotics for a nontrivial MVOP class with a solid core argument; the printed Theorem 1.3 has a domain-of-definition flaw that needs fixing before the results are citable as stated. read the letter →

arxiv 2412.03115 v1 pith:VWYF2FRY submitted 2024-12-04 math.CA math-phmath.MP

Arno B.J. Kuijlaars This is my paper · ORCID
classification math.CAmath-phmath.MP MSC 42C0530E2533E0541A60
keywords matrixvaluedorthogonalpolynomialsdoublyperiodictilingslozengeofahexagonRiemann-HilbertproblemsteepestdescentasymptoticsequilibriummeasureHarnackcurvesgenusonespectralcurve
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Matrix-valued orthogonal polynomials (MVOP) are polynomials with matrix coefficients that are orthogonal with respect to a matrix weight on a contour; they encode the correlation functions of doubly periodic lozenge tilings of a hexagon. This paper studies the special 3x3-periodic weights satisfying the symmetry conditions (1.4)--(1.7) in the regular-hexagon case $B=C=1$, and proves a strong asymptotic formula for the degree-$N$ MVOP $P_N$: for $z$ away from the unit circle, $P_N(z)=L^N(A_N(z)+O(e^{-cN}/(1+|z|)))G(z)^N$ with an explicit matrix $G$ built from the $g$-functions of the spectral curve. The central result is that the normalized zero counting measures of $\det P_N$ converge weakly, as $N\to\infty$, to a probability measure on the unit circle with a positive real-analytic density, identified explicitly as the balayage (harmonic sweep) of $\delta_{P_\infty}-\delta_{P_1}+\delta_{P_0}$ onto the first two sheets' unit circles. If the paper is right, this supplies a complete large-$N$ analysis of MVOP from hexagon tilings in a genuinely two-parameter family, and it exhibits the equilibrium measure with the S-property in the external field $\mathrm{Re}\,V=2\log|z|-2\log|\lambda|$ as the organizing principle.

What carries the argument

The machinery is organized around the genus-one spectral curve $P(z,\lambda)=(\lambda-z-1)^3-27(1+\beta)\lambda z=0$, a Harnack curve (an algebraic curve whose logarithmic amoeba map is at most two-to-one) with holomorphic symmetries $(z,\lambda)\mapsto(-\lambda,-z)$ and $(z,\lambda)\mapsto(z^{-1},\lambda/z)$ plus complex conjugation. The strict ordering $|\lambda_1(z)|>|\lambda_2(z)|>|\lambda_3(z)|>0$ off the real axis selects three sheets and defines the eigenvector matrix $E(z)$. The Abel map sends this curve to the complex torus $\mathbb{C}/(\mathbb{Z}+\tau\mathbb{Z})$, with the three special points $P_\infty,P_1,P_0$ at $1/6,1/2,5/6$ and the circles $\Gamma_1\cup\Gamma_2$, $\Gamma_3$ on the two halves of the $b$-cycle. The candidate limiting measure is $\mu=\mathrm{Bal}(\delta_{P_\infty}-\delta_{P_1}+\delta_{P_0};\Gamma_1\cup\Gamma_2)$; its positivity is verified by transporting it to a vertical strip and then to the upper half-plane, where it becomes a sum of three Cauchy densities that is positive. The $g$-functions are defined from the complexified bipolar Green's kernel $G^C_{P_\infty}$, written with Jacobi $\theta$ functions $\theta_1$, and satisfy the jump and normalization conditions that make the Riemann-Hilbert transformations work. The global parametrix is built row by row from meromorphic functions $\psi_j,\phi_j$ constructed from Jacobi $\theta$ functions, with prescribed poles at $Q^*_j$ and zeros, arranged as Hadamard products $E\circ\Psi$ and $E\circ\Phi$, with a constant matrix $K$ fixing the behavior at infinity; parity of $N$ selects one of two such constructions. The final transformation $S\to R$ reduces to a small-norm Riemann-Hilbert problem, giving $R=I_6+O(e^{-cN})$ and hence the exponential error in Theorem 1.3.

What would settle it

Take the explicit model (1.8) with $\alpha_1=0.2$, $\alpha_2=2$, compute the zeros of $\det P_N$ numerically for large $N$ (say $N=100$ and $N=200$), and compare the empirical normalized counting measure with the pushforward $\mu_*=(\pi_z)_*\mu$ of $\mu=\mathrm{Bal}(\delta_{P_\infty}-\delta_{P_1}+\delta_{P_0};\Gamma_1\cup\Gamma_2)$; a mismatch, or any zeros accumulating away from the unit circle, would refute Theorems 1.3--1.4. A second check is to verify the normal-derivative identity (1.41) for the harmonic function $h=2U^\mu+\mathrm{Re}\,V-\ell$ using the $\theta$-function formulas from Section 4.

Watch

Extended reading notes

Core claim

The paper's central discovery is that, under the symmetry assumptions (1.4)--(1.7) with $B=C=1$, the monic matrix-valued orthogonal polynomial $P_N$ associated with the 3x3-periodic hexagon model has the strong asymptotic expansion $P_N(z)=L^N(A_N(z)+O(e^{-cN}/(1+|z|)))G(z)^N$ uniformly for $z$ in compact subsets of $\mathbb{C}\setminus\mathbb{T}$, where $L$ is unit lower triangular, $A_N$ depends only on the parity of $N$, and $G(z)=E(z)\operatorname{diag}(e^{g_1(z)},e^{g_2(z)},e^{g_3(z)})E(z)^{-1}$; the $g_j$ are normalized so that $g_j(z)=\log z+O(z^{-1/3})$ as $z\to\infty$. From this formula the author derives that the normalized zero counting measures of $\det P_N$ converge weakly to $\mu_*=(\pi_z)_*\mu$, where $\mu=\mathrm{Bal}(\delta_{P_\infty}-\delta_{P_1}+\delta_{P_0};\Gamma_1\cup\Gamma_2)$ is supported on the two unit circles on sheets 1 and 2 and has a positive real-analytic density. The measure $\mu$ is characterized by the harmonic identity $\int h\,d\mu=h(P_\infty)-h(P_1)+h(P_0)$ for functions harmonic off $\Gamma_1\cup\Gamma_2$, and it is shown to be the equilibrium measure of $\Gamma_1\cup\Gamma_2\cup\Gamma_3$ in the external field $\mathrm{Re}\,V$ with the S-property: $2U^\mu+\mathrm{Re}\,V=\ell$ on $\Gamma_1\cup\Gamma_2$ and $2U^\mu+\mathrm{Re}\,V>\ell$ on $\Gamma_3$. The spectral curve behind all this is the explicit genus-one Harnack curve $(\lambda-z-1)^3=27(1+\beta)\lambda z$ with $\beta>0$, whose three special points $P_0,P_\infty,P_1$ sit at $z=0$, $z=\infty$, and $(-1,0)$ respectively.

Load-bearing premise

The argument rests on the imported theorem, not reproved in the paper, that the doubly periodic weight matrix $W(z)$ has a Harnack spectral curve whose eigenvalues obey the strict ordering $|\lambda_1(z)|>|\lambda_2(z)|>|\lambda_3(z)|>0$ for $z\notin\mathbb{R}$; all sheet labels, the eigenvector matrix $E(z)$, and the first Riemann-Hilbert transformation depend on that ordering.

Editorial extensions

If this is right

  • For the model (1.8), the zeros of $\det P_N$ cluster on the unit circle with a density that can be computed from the parameter $\beta$ alone, so the asymptotic zero distribution is completely determined by the single parameter $\beta$ (equivalently, by $\alpha_1,\alpha_2$).
  • The strong formula (1.18) determines $\det P_N$ up to exponentially small relative error on compact sets away from the unit circle; in particular, no zeros accumulate outside $\mathbb{T}$ and the convergence in Theorem 1.4 is exponentially fast away from the circle.
  • Because $\mu$ has full support on $\Gamma_1\cup\Gamma_2$ with a positive analytic density, the lens-opening step does not require local parametrices, which is why the error term in (1.18) is exponentially small rather than merely $O(N^{-1})$.
  • The parity dependence of $A_N$ and the explicit jump relations (1.24)--(1.25) imply that the even- and odd-$N$ polynomials have different leading analytic factors but share the same limiting zero measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves general $B,C$ open; following Remark 1.8, for $B=C$ the equality in the Euler-Lagrange conditions on $\Gamma_1\cup\Gamma_2$ persists with the generalized balayage (1.43), so a natural test is whether positivity, the strict inequality on $\Gamma_3$, and the S-property hold for all $0<C\le 1$, which would extend the strong asymptotics beyond the regular hexagon.
  • Because the same Riemann-Hilbert solution enters the correlation kernel (A.4), the explicit equilibrium measure should determine the rough-phase geometry and the arctic curve of the random tiling model; the paper states the derivation of these tiling consequences as a future step.
  • The same scheme---an explicit genus-one spectral curve, the Abel map to a torus, a theta-function global parametrix, and a full-support equilibrium measure---is the natural template for higher periodicities with analogous symmetries, with the three-point balayage replaced by the divisor supplied by Abel's theorem.
  • The explicit formula for $\mu$ implies the limiting zero density is invariant under complex conjugation $z\mapsto\bar{z}$ (equivalently $\theta\mapsto-\theta$ on the unit circle), a symmetry visible in the $N=20$ plot and directly checkable for larger $N$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper analyzes matrix-valued orthogonal polynomials (MVOP) associated with 3x3-periodic lozenge tilings of a regular hexagon, in the special case B=C=1 under the symmetry assumptions (1.4)-(1.7). The spectral curve is reduced to the explicit genus-one curve (2.1), with branch points zmin and zmax. The authors define a probability measure mu as the balayage of delta_{P_infty} - delta_{P_1} + delta_{P_0} onto Gamma_1 union Gamma_2, prove that it is positive with real-analytic density, verify the variational equalities and the S-property, and use the associated g-functions in a Deift-Zhou steepest descent analysis of the Riemann-Hilbert problem. The main results are a strong asymptotic formula for the MVOP P_N (Theorem 1.3) and the weak convergence of the normalized zero counting measures of det P_N to the pushforward of mu onto the unit circle (Theorem 1.4). The appendix connects the MVOP to correlation kernels of random lozenge tilings.

Significance. If the stated results hold, this is a substantial advance: it provides the first full steepest-descent analysis for MVOP arising from doubly periodic lozenge tilings with period three, with an explicitly identified genus-one equilibrium measure and exponential error estimates. The paper is unusually concrete: the spectral curve is explicit, the positivity of the balayage measure is verified by an elementary calculation, the g-functions are defined from the equilibrium measure rather than fitted, and the global parametrix is built explicitly from Jacobi theta functions. The absence of local parametrices and the exponential smallness of the remainders are notable strengths. The main claims are falsifiable and the proofs are laid out in enough detail that the individual steps can be checked. The central issue identified below concerns the precise domain on which the strong asymptotic formula is stated, and a gap in one step of the proof of the zero-distribution theorem; both appear to be repairable without changing the overall strategy.

major comments (2)
  1. [Section 1.3, Theorem 1.3 and Eq. (1.18)] The statement says (1.18) holds uniformly for z in compact subsets of C\T, but by part (a) G is defined and analytic only on C\(T ∪ [zmin,zmax]), and part (d) gives a jump across [zmin,1)∪(1,zmax]. Since Assumption 1.2(d) gives zmin<1<zmax, the cut lies entirely in C\T, so an admissible compact set may intersect it; at such a point G(z)^N has no single value, and for odd N the two boundary limits differ by conjugation with diag(1,-1,-1). The derivation in Section 10.1 proves (1.18) only outside the lens around T, and choosing the lens arbitrarily thin does not repair the problem, because cut points are separated from T but are not in the domain of G. The theorem should either restrict to compact subsets of C\(T ∪ [zmin,zmax]) or specify one-sided boundary values on the cut, with the compensation by A_N understood via (1.24)-(1.25).
  2. [Section 10.3, proof of Theorem 1.4(c)] The argument that det A_e and det A_o are not identically zero inside the unit circle is not fully justified. First, (1.26) is stated uniformly for compact subsets of C\T, so it does not directly give the bound sup_{|z|≤1}|det P_N(z)| ≤ C e^{-cN} used in (10.5), because the closed unit disk meets T. Second, for odd N the function A_o is not analytic in C\T: by Theorem 1.3(c) it is analytic in C\(T ∪ [zmin,zmax]) and has the jump (1.25) across the cut, which splits the interior of the unit disk. The proof needs a boundary-value or maximum-principle argument on annuli or domains with boundary arcs, or a different argument, before one can conclude that the only zeros accumulating on T are those captured by the asymptotic formula.
minor comments (4)
  1. [Section 8.2, paragraph after (8.8)] The sentence 'Thus there is k ∈ {2,3} such that E_{j,k}(z_j^*)' is incomplete; it should say that E_{j,k}(z_j^*) = 0.
  2. [Section 10.3] The expression 'sup_{z∈T} det |P_N(z)|' should read 'sup_{z∈T} |det P_N(z)|'.
  3. [Section 10.1, last paragraph] The sentence 'Since η > 0 can be taken arbitrarily close to 0' should be made precise by referencing the positive distance from a given compact set to T; as written it can be read as claiming uniformity over all of C\T, which interacts with the domain issue raised in Major Comment 1.
  4. [Lemma 6.3(b)] The same symbol k1 is used in (6.8) and (6.9), while (6.10) uses k2; this is confusing because the three equations refer to three different connected components and the constants need not coincide. Renaming the constants would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equilibrium measure is constructed explicitly and verified against variational conditions, not fitted to the zeros; self-citations supply context and existence, not the asymptotic conclusion.

full rationale

The central asymptotic theorem (Theorem 1.3) is derived from a Deift-Zhou steepest descent analysis of the Riemann-Hilbert problem 1.10, not from a fitted ansatz. The equilibrium measure is not fitted: it is defined in (1.31) as the balayage Bal(delta_{P_infty} - delta_{P_1} + delta_{P_0}; Gamma_1 union Gamma_2), and Theorem 1.7 then proves in the body of the paper that this explicit candidate satisfies the Euler-Lagrange equality on Gamma_1 union Gamma_2, strict inequality on Gamma_3, and the S-property (1.40)-(1.41). The g-functions are defined from this measure in (6.5) and (6.17), and the zero-counting limit in Theorem 1.4(c) follows from the resulting strong asymptotics together with standard potential-theoretic uniqueness from Saff-Totik, not from assuming the conclusion. The principal spectral-curve input, the Harnack property giving the eigenvalue ordering (1.10), is imported from Kenyon-Okounkov-Sheffield, which is external to the author, and is not replaced by a self-citation. The author's own prior works [10], [32], and [37] are used for context, for the MVOP-tiling connection, and for existence/uniqueness of the MVOP; these are premises rather than the load-bearing asymptotic argument, and the asymptotic content itself is worked out from the RH analysis in Sections 5-10. No fitted parameters are renamed as predictions, and no equation is exhibited that reduces to its own input by construction. A separate non-circularity concern is present in the statement of Theorem 1.3: it claims uniformity on compact subsets of C \ T while G is defined only on C \ (T union [zmin,zmax]) and has a jump across [zmin,1) union (1,zmax]; this is a correctness and well-posedness issue, not a circularity. For circularity purposes, the derivation is self-contained once the external Harnack-curve theorem and the prior MVOP existence results are granted.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim depends on standard results from earlier literature: existence and uniqueness of the MVOP, Harnack curve structure, bipolar Green's kernel and theta function identities, and Abel's theorem. None of these are invented for this paper. The only computational step unique to the paper is the Maple-assisted simplification in Lemma 2.1, which is not shown by hand. There are no fitted numbers and no invented entities.

assumptions (5)
  • standard math Existence and uniqueness of the MVOP with non-hermitian orthogonality (1.3), cited from Duits-Kuijlaars [32] and Groot-Kuijlaars [37].
    The paper does not prove existence or uniqueness of P_N and relies on prior theorems for the tiling connection.
  • standard math Harnack curve property for the spectral curve, giving strict eigenvalue ordering (1.10) and the sheet structure of Figure 2, cited from Kenyon-Okounkov [41] and Kenyon-Okounkov-Sheffield [42].
    This external theorem underpins the definition of the three sheets and the eigenvector matrix E(z), which is load-bearing for the RH analysis.
  • standard math Bipolar Green's kernel existence, uniqueness up to constant, and the theta function representation (1.36), cited from Skinner [53] and Kang-Makarov [40].
    The equilibrium measure and g-functions are defined through this kernel, and the complexified form in Definition 6.1 uses the theta formulas.
  • standard math Abel's theorem and standard quasi-periodicity properties of the Jacobi theta function are used in Proposition 2.4 and in the global parametrix construction.
    The Abel map computations and the meromorphic function constructions in Lemmas 8.4 and 8.7 depend on these classical facts.
  • ad hoc to paper The simplification Q=(A+B+C)^3/(ABC) in Lemma 2.1 is correct.
    The paper states 'Assisted by Maple' and gives no hand derivation or code, so this is the least independently verified computational step in the derivation of the spectral curve equation.

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Pith. "Pith review of Matrix valued orthogonal polynomials arising from hexagon tilings with 3x3-periodic weightings." pith.science (2026). https://pith.science/paper/VWYF2FRY

@misc{pith2026241203115,
  author       = {Pith},
  title        = {Pith review of: Matrix valued orthogonal polynomials arising from hexagon tilings with 3x3-periodic weightings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWYF2FRY}},
  note         = {Machine review of arXiv:2412.03115}
}
read the original abstract

Matrix valued orthogonal polynomials (MVOP) appear in the study of doubly periodic tiling models. Of particular interest is their limiting behavior as the degree tends to infinity. In recent years, MVOP associated with doubly periodic domino tilings of the Aztec diamond have been successfully analyzed. The MVOP related to doubly periodic lozenge tilings of a hexagon are more complicated. In this paper we focus on a special subclass of hexagon tilings with 3x3 periodicity. The special subclass leads to a genus one spectral curve with additional symmetries that allow us to find an equilibrium measure in an external field explicitly. The equilibrium measure gives the asymptotic distribution for the zeros of the determinant of the MVOP. The associated g-functions appear in the strong asymptotic formula for the MVOP that we obtain from a steepest descent analysis of the Riemann-Hilbert problem for MVOP.

Figures

Figures reproduced from arXiv: 2412.03115 by the authors.

Figure 1
Figure 1. Sheet structure of the Riemann surface for generic parameters. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 1
Figure 1. If z4 = z5 then the bounded oval degenerates to a node. In that case the Riemann surface has genus zero, otherwise the genus is one. The assumptions for the present paper are the following. Assumptions 1.2. We assume (a) 0 is a branch point of the Riemann surface that connects all three sheets and λk(z) → 1 as z → 0 for every k = 1, 2, 3; (b) ∞ is a branch point of the Riemann surface that connects all three sheets … view at source ↗
Figure 2
Figure 2. Sheet structure for the Riemann surface under Assumptions 1.2. The [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figures from the paper (15 more)
Figure 2
Figure 2. Figure 2: (b) Let µ∗ = (πz)∗(µ) be the pushforward measure of µ under the projection map (1.28). Then µ∗ is a probability measure on T with the property that Re (g1(z) + g2(z) + g3(z)) = 3 Z log |z − s|dµ∗(s), z ∈ C \ T. (1.30) (c) Let PN be the MVOP satisfying the matrix orthog…
Figure 3
Figure 3. Figure 3: Zeros of det PN with N = 20 coming from the model (1.8) with parameters α1 = 0.2, α2 = 2. The zeros tend to the unit circle, in the sense of weak convergence of normalized zero counting measures, by Theorem 1.4. as, for any p ∈ R, the balayage measure (also known as ha…
Figure 4
Figure 4. Figure 4: The unit circles Γ1, Γ2 and Γ3 on the three sheets of the Riemann surface. Theorem 1.7. Let µ be the probability measure from Theorem 1.4 and let V be given by (1.33). Then there is a constant ℓ such that 2U µ + Re V ( = ℓ, on Γ1 ∪ Γ2, > ℓ, on Γ3, (1.40) and µ is the e…
Figure 5
Figure 5. Figure 5: Complex torus C/(Z + τZ) and the image of the Riemann surface under the Abel map (2.14). Proof. (a) Since a is fixed under the anti-holomorphic involution (2.7) the holomor￾phic differential ω will be of the form ω = vdz with a meromorphic v that is real and positive o…
Figure 5
Figure 5. Figure 5: At these special points there are logarithmic singularities [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Real part of the spectral curve in the z-λ plane (in black) together with the zero level lines of the functions that define the matrix E. The three zero level lines intersect the spectral curve at the point z = z ∗ , λ = λ ∗ on the bounded oval. Each zero level line ha…
Figure 7
Figure 7. Figure 7: Contour ΣX for the RH problem 5.6 for X. Recall that we use the bipolar Green’s kernel as in Definition 1.6 with the ad￾ditional normalization (4.1) at infinity. Thus in view of (1.36) and the fact that A(P∞) ≡ 1 6 , we have with u = A(p), v = A(q), GP∞(p, q) = log [P…
Figure 8
Figure 8. Figure 8: Contour ΣS = R ∪ {|z| = 1} ∪ {|z| = 1 ± η} for the RH problem 7.2 for S. RH problem 7.2. S satisfies the following RH problem. RHP-S1 S : C \ ΣS → C 6×6 is analytic, where ΣS = R ∪ T ∪ {|z| = 1 ± η}. RHP-S2 S+ = S−JS on ΣS where JS = Π  0 (−1)N −(−1)N 0  ⊕  0 (−1)N…
Figure 9
Figure 9. Figure 9: Construction of Ψ and Φ in case N is even. For j = 1, 2, 3 there is a meromorphic function fj on the rectangle (0, 2) × (0, τ ) with poles at xj + τ 2 and xj + 1 + τ 2 , zeros at 7 6 and 2xj − 1 6 + τ 2 , and normalization such that fj (1/6) = 1. The jth row of Ψ is fi…
Figure 10
Figure 10. Figure 10: Construction of Ψ and Φ in case N is odd. For j = 1, 2, 3 there is a meromorphic function fj on the rectangle (0, 2) × (− τ 2 , τ 2 ) with poles at xj + τ 2 and xj + 1 + τ 2 , zeros at 1 6 and 2xj − 5 6 + τ 2 , and normalization such that fj (7/6) = 1. The jth row of …
Figure 11
Figure 11. Figure 11: Contour ΣR = T ∪ {|z| = 1 ± η} for the RH problem 9.2 for R. 9 Final transformation The final transformation is Definition 9.1. We define for z ∈ C \ ΣS, R(z) = S(z)M(z) −1 . (9.1) With the definition (9.1) we restore the property that det R = 1. We have the following…
Figure 12
Figure 12. Figure 12: A tiling is equivalent to a family of non-intersecting paths going from [PITH_FULL_IMAGE:figures/full_fig_p052_12.png]
Figure 12
Figure 12. Figure 12: Lozenge tiling of a hexagon left to right that follow the blue and red lozenges (in the coloring of [PITH_FULL_IMAGE:figures/full_fig_p053_12.png]
Figure 13
Figure 13. Figure 13: Non-intersecting paths and 3 × 3 doubly periodic weights on the tiling from Figure (12). for every n, m ∈ Z. See [PITH_FULL_IMAGE:figures/full_fig_p054_13.png]
Figure 14
Figure 14. Figure 14: Random tiling of a hexagon with 3 × 3 doubly periodic weights of the form considered in this paper. The parameters are N = 200 and α1 = α2 = 0.3, see (1.8). The model exhibits three asymptotic phases: solid, rough and smooth. The smooth phase in the middle has six cus…

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Works this paper leans on

55 extracted references · 52 canonical work pages

  1. [1]

    The P\'olya-Tchebotarev problem with semiclassical external fields

    V. Alves and G.L.F. Silva, The P´ olya-Tchebotarev problem with semiclassical external fields, arXiv:2403.00719. 56

    work page Pith review arXiv

  2. [2]

    Bain, Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit, J

    E. Bain, Local correlation functions of the two-periodic weighted Aztec diamond in mesoscopic limit, J. Math. Phys. 64 (2023), paper 023301, 64 pp

    work page 2023

  3. [3]

    Beffara, S

    V. Beffara, S. Chhita, K. Johansson, Airy point process at the liquid-gas bound- ary, Ann. Probab. 46 (2018), 2973–3013

    work page 2018

  4. [4]

    Berggren, Domino tilings of the Aztec diamond with doubly periodic weight- ings, Ann

    T. Berggren, Domino tilings of the Aztec diamond with doubly periodic weight- ings, Ann. Probab. 49 (2021), 1965–2011

    work page 2021

  5. [5]

    Berggren and A

    T. Berggren and A. Borodin, Geometry of the doubly periodic Aztec dimer model, arXiv:2306.07482

    arXiv

  6. [6]

    Berggren and M

    T. Berggren and M. Duits, Correlation functions for determinantal processes defined by infinite block Toeplitz minors, Adv. Math. 356 (2019) 106766

    work page 2019

  7. [7]

    Bertola, Pad´ e approximants on Riemann surfaces and KP tau functions, Anal

    M. Bertola, Pad´ e approximants on Riemann surfaces and KP tau functions, Anal. Math. Phys. 11 (2021), no. 4, Paper No. 149, 38 pp

    work page 2021

  8. [8]

    Bertola, Nonlinear steepest descent approach to orthogonality on elliptic curves, J

    M. Bertola, Nonlinear steepest descent approach to orthogonality on elliptic curves, J. Approx. Theory 276 (2022), Paper No. 105717, 33 pp

    work page 2022

Show all 55 references
  1. [9]

    Bertola, Abelianization of matrix orthogonal polynomials, Int

    M. Bertola, Abelianization of matrix orthogonal polynomials, Int. Math. Res. Not. IMRN 2023 (2023), no. 10, 8544–8595

    2023

  2. [10]

    Bertola, A

    M. Bertola, A. Groot, and A.B.J. Kuijlaars, Critical measures on higher genus Riemann surfaces, Comm. Math. Phys. 404 (2023), 51–95

    2023

  3. [11]

    Bleher and A

    P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. Math. 150 (1999) 185–266

    1999

  4. [12]

    Bobenko and N

    A.I. Bobenko and N. Bobenko, Dimers and M -curves: limit shapes from Rie- mann surfaces, arXiv:2407.19462

    arXiv

  5. [13]

    Borodin and M

    A. Borodin and M. Duits, Biased 2 × 2 periodic Aztec diamond and an elliptic curve, Prob. Theory Rel. Fields 187 (2023), 259–315

    2023

  6. [14]

    Boutillier, D

    C. Boutillier, D. Cimasoni, and B. de Tilli` ere, Minimal bipartite dimers and higher genus Harnack curves, Probab. Math. Phys. 4 (2023), 151–208

    2023

  7. [15]

    Cassatella-Contra and M

    G.A. Cassatella-Contra and M. Ma˜ nas, Riemann-Hilbert problems, matrix or- thogonal polynomials and discrete matrix equations with singularity confine- ment, Stud. Appl. Math. 128 (2012), 252–274

    2012

  8. [16]

    Charlier, Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials, Stud

    C. Charlier, Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials, Stud. Appl. Math. 146 (2021), 3–80

    2021

  9. [17]

    Charlier, Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface, Trans

    C. Charlier, Matrix orthogonality in the plane versus scalar orthogonality in a Riemann surface, Trans. Math. Appl. 5 (2021), tnab004, 35 pp. 57

    2021

  10. [18]

    Charlier, M

    C. Charlier, M. Duits, A.B.J. Kuijlaars, and J. Lenells, A periodic hexagon tiling model and non-hermitian orthogonal polynomials, Comm. Math. Phys. 378 (2020), 401–466

    2020

  11. [19]

    Chhita and M

    S. Chhita and M. Duits, On the domino shuffle and matrix refactorizations, Comm. Math. Phys. 401 (2023), 1417–1467

    2023

  12. [20]

    Chhita and K

    S. Chhita and K. Johansson, Domino statistics of the two-periodic Aztec dia- mond, Adv. Math. 294 (2016), 37–149

    2016

  13. [21]

    Chirka, Potentials on a compact Riemann surface, Proc

    E.M. Chirka, Potentials on a compact Riemann surface, Proc. Steklov Inst. Math. 301 (2018), 272–303

    2018

  14. [22]

    Chirka, Equilibrium measures on a compact Riemann surface, Proc

    E.M. Chirka, Equilibrium measures on a compact Riemann surface, Proc. Steklov Inst. Math. 306 (2019), 296–334

    2019

  15. [23]

    Chirka, Capacities on a compact Riemann surface, Proc

    E.M. Chirka, Capacities on a compact Riemann surface, Proc. Steklov Inst. Math. 311 (2020), 36–77

    2020

  16. [24]

    Damanik, A

    D. Damanik, A. Pushnitski, and B. Simon, The analytic theory of matrix or- thogonal polynomials, Surv. Approx. Theory 4 (2008) 1–85

    2008

  17. [25]

    Dea˜ no, A.B.J

    A. Dea˜ no, A.B.J. Kuijlaars, and P. Rom´ an, Asymptotics of matrix valued or- thogonal polynomials on [ −1, 1], Adv. Math. 423 (2023), Paper 109043, 61 pp

    2023

  18. [26]

    Deift, Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach, Courant Lecture Notes in Mathematics 3

    P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach, Courant Lecture Notes in Mathematics 3. Amer. Math. Soc, Provi- dence, RI, 1999

    1999

  19. [27]

    Deift, T

    P. Deift, T. Kriecherbauer, K.T-R McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying ex- ponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425

    1999

  20. [28]

    Deift and X

    P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann- Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. 137 (1993), 295–368

    1993

  21. [29]

    Delvaux, Average characteristic polynomials for multiple orthogonal polyno- mial ensembles, J

    S. Delvaux, Average characteristic polynomials for multiple orthogonal polyno- mial ensembles, J. Approx. Theory 162 (2010), 1033–1067

    2010

  22. [30]

    Desiraju, A.R

    H. Desiraju, A.R. Its, and A. Prokhorov, Nonlinear steepest descent on a torus: A case study of the Landau-Lifshitz equation, preprint arXiv:2405.17662

    arXiv

  23. [31]

    Desiraju, T.L

    H. Desiraju, T.L. Latimer, and P. Roffelsen, On a class of elliptic orthogonal polynomials and their integrability, to appear in Constructive Approximation, preprint arXiv:2305.04404

    arXiv

  24. [32]

    Duits and A.B.J

    M. Duits and A.B.J. Kuijlaars, The two-periodic Aztec diamond and matrix valued orthogonal polynomials, J. Eur. Math. Soc. 23 (2021), 1075–1131. 58

    2021

  25. [33]

    Eynard and M.L

    B. Eynard and M.L. Mehta, Matrices coupled in a chain. I. Eigenvalue correla- tions. J. Phys. A 31 (1998), 4449–4456

    1998

  26. [34]

    Fasondini, S

    M. Fasondini, S. Olver, and Y. Xu, Orthogonal polynomials on planar cubic curves, Found. Comput. Math. 23 (2023), 1–31

    2023

  27. [35]

    Fokas, A.R

    A.S. Fokas, A.R. Its, and A.V. Kitaev, The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395–430

    1992

  28. [36]

    Gessel and G

    I. Gessel and G. Viennot, Binomial determinants, paths, and hook length for- mulae. Adv. Math. 58 (1985), 300–321

    1985

  29. [37]

    Groot and A.B.J

    A. Groot and A.B.J. Kuijlaars, Matrix-valued orthogonal polynomials related to hexagon tilings, J. Approx. Theory 270 (2021), 105619, 36 pp

    2021

  30. [38]

    Gr¨ unbaum, M.D

    F.A. Gr¨ unbaum, M.D. de la Iglesia, and A. Mart´ nez-Finkelshtein, Properties of matrix orthogonal polynomials via their Riemann-Hilbert characterization, SIGMA Symmetry Integr. Geom. Methods Appl. 7 (2011) 098, 31 pages

    2011

  31. [39]

    Johansson and S

    K. Johansson and S. Mason, Dimer-dimer correlations at the the rough-smooth boundary, Comm. Math. Phys. 400 (2023), 1255–1315

    2023

  32. [40]

    Kang and N

    N.-G. Kang and N. Makarov, Calculus of conformal fields on a compact Rie- mann surface, arXiv:1708.07361, 86 pp

    arXiv

  33. [41]

    Kenyon and A

    R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, Duke Math. J. 131 (2006), 499–524

    2006

  34. [42]

    Kenyon, A

    R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae, Ann. Math. 163 (2006), 1019–1056

    2006

  35. [43]

    Kuijlaars and M

    A.B.J. Kuijlaars and M. Piorkowski, Wiener-Hopf factorization and matrix- valued orthogonal polynomials, arXiv:2402.07706

    arXiv

  36. [44]

    Kuijlaars and G.L.F

    A.B.J. Kuijlaars and G.L.F. Silva, S-curves in polynomial external fields, J. Approx. Theory 191 (2015), 1–37

    2015

  37. [45]

    Lindstr¨ om, On the vector representations of induced matroids, Bull

    B. Lindstr¨ om, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85–90

    1973

  38. [46]

    Mart ´ ınez-Finkelshtein, K.T.-R

    A. Mart ´ ınez-Finkelshtein, K.T.-R. McLaughlin and E.B. Saff, Szeg˝ o orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics, Constr. Approx. 24 (2006), 319–363

    2006

  39. [47]

    Mart ´ ınez-Finkelshtein and E.A

    A. Mart ´ ınez-Finkelshtein and E.A. Rakhmanov, Critical measures, quadratic differentials and weak limits of zeros of Stieltjes polynomials, Comm. Math. Phys. 302 (2011), 53–111

    2011

  40. [48]

    Mart ´ ınez-Finkelshtein and E.A

    A. Mart ´ ınez-Finkelshtein and E.A. Rakhmanov, Do orthogonal polynomials dream of symmetric curves? Found. Comput. Math. 16 (2016), 1697–1736. 59

    2016

  41. [49]

    Mart ´ ınez-Finkelshtein and G.L.F

    A. Mart ´ ınez-Finkelshtein and G.L.F. Silva, Critical measures for vector energy: global structure of trajectories of quadratic differentials, Adv. Math. 302 (2016), 1137–1232

    2016

  42. [50]

    Piorkowski, Arctic curves of periodic dimer models and generalized discrim- inants, arXiv:2410.17138

    M. Piorkowski, Arctic curves of periodic dimer models and generalized discrim- inants, arXiv:2410.17138

    arXiv

  43. [51]

    Rakhmanov, Orthogonal polynomials and S-curves, in: Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Con- temp

    E.A. Rakhmanov, Orthogonal polynomials and S-curves, in: Recent Advances in Orthogonal Polynomials, Special Functions, and their Applications, Con- temp. Math. 578, Amer.Math. Soc., Providence RI, pp. 195–239, 2012

    2012

  44. [52]

    Saff and V

    E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, in: Grundlehren der mathematischen Wissenschaften, vol. 316, Springer-Verlag, Berlin, 1997, second edition Springer Nature Switzerland, 2024

    1997

  45. [53]

    B. Skinner, Logarithmic Potential Theory on Riemann Surfaces,Dissertation (Ph.D.), California Institute of Technology, doi:10.7907/Z9Q52MK8, https://resolver.caltech.edu/CaltechTHESIS:05292015-072640484

    doi

  46. [54]

    Stahl, Extremal domains associated with an analytic function

    H. Stahl, Extremal domains associated with an analytic function. I, II. Complex Variables Theory Appl. 4 (1985), 311–324, 325–338

    1985

  47. [55]

    Stahl, Orthogonal polynomials with complex-valued weight function

    H. Stahl, Orthogonal polynomials with complex-valued weight function. I, II. Constr. Approx. 2 (1986), 225–240, 241–251. 60

    1986

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