arXiv subjects

Maurice Duits

Publications and source records attributed to Maurice Duits.

At least 19 recordsLinked to original sources

arXiv2026

On Toeplitz determinants with slow Fourier decay

We study Toeplitz determinants $\det T_n(e^f)$ for $f$ whose Fourier coefficients satisfy $f_k=O(|k|^{-1})$. This regime extends beyond $H^{1/2}$ and includes symbols with Fisher-Hartwig singularities. We develop an operator-theoretic approach based on the Baker-Campbell-Hausdorff formula that separates the quadratic term \[ \sum_{k=1}^{\infty}\min(k,n)f_kf_{-k} \] from the higher-order terms in the expansion of $\log\det T_n(e^{tf})$. We show that this quadratic term accounts for the possible growth with $n$, while every fixed higher-order coefficient remains bounded. For symbols with bounded positive and negative Fourier parts, our estimates yield two-sided bounds for the determinant after removal of the quadratic contribution. For a broader admissible class, including Fisher-Hartwig-type symbols, we obtain uniform higher-order coefficient bounds and a central limit theorem for the associated CUE linear statistics. We also obtain bounds on mixed exponential moments for CUE-derived random fields beyond the characteristic polynomial.

math-ph↗

arXiv2026

The Ising Model Coupled to 2D Gravity: Critical Partition Function

We prove that the differential of the log of the partition function for the $2$-matrix model with quartic interactions converges in a certain double-scaling regime to the differential of the $\boldsymbol{\tau}$-function for the $(3,4)$ string equation. This confirms the convergence of the critical Ising model on random surfaces to the $(3,4)$ topological minimal model, which was stated in the works of Douglas and Shenker, Br\'{e}zin and Kazakov, and Gross and Migdal. Our analysis is based on a steepest-descent analysis of a Riemann-Hilbert problem associated to a family of biorthogonal polynomials. New features in the matching problem in the construction of local parametrices appear.

math-ph↗

arXiv2025

The Gamma-disordered Aztec diamond

We introduce a multi-parameter family of random edge weights on the Aztec diamond graph, given by certain Gamma variables, and prove several results about the corresponding random dimer measures. Firstly, we show there is no phase transition at the level of the free energy. This provides rigorous backing for the physics predictions of Zeng-Leath-Hwa and later works that dimer models with random weights are in the glassy `super-rough' phase at all temperatures with no phase transition. Secondly, we show that the random dimer covers themselves enjoy exact distributional equalities of certain marginals with path locations in new `hybrid' integrable polymers. These reduce to the stationary log-Gamma, strict-weak, and Beta polymer in random environment in certain cases, allowing transfer of known results from integrable polymers to dimers with random weights. As an example application, we prove that the turning points at the boundaries of the Aztec diamond exhibit fluctuations of order $n^{2/3}$, in contrast to the $n^{1/2}$ fluctuations for deterministic weights. Underlying all these is a key integrability property of the weights: they are the unique family for which independence is preserved under the shuffling algorithm.

math.PR↗

arXiv2025

The $q^{\mathrm{Volume}}$ lozenge tiling model via non-Hermitian orthogonal polynomials

We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of non-intersecting paths. The starting point of our analysis is a formula for the correlation kernel due to Duits and Kuijlaars which involves the Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal polynomials. Our main results are split into two parts: the first part concerns the family of orthogonal polynomials, and the second concerns the behavior of the boundary of the so-called arctic curve. In the first half, we identify the orthogonal polynomials as a non-standard instance of little $q$-Jacobi polynomials and compute their large degree asymptotics in the $q \to 1$ regime. A consequence of this analysis is a proof that the zeros of the orthogonal polynomials accumulate on an arc of a circle and an asymptotic formula for the Christoffel-Darboux kernel. In the second half, we use these asymptotics to show that the boundary of the liquid region converges to the Airy process, in the sense of finite dimensional distributions, away from the boundary of the hexagon. At inflection points of the arctic curve, we show that we do not need to subtract/add a parabola to the Airy line ensemble, and this effect persists at distances which are $o(N^{-2/9})$ in the tangent direction.

math-ph↗

arXiv2024

The Ising Model Coupled to 2D Gravity: Genus Zero Partition Function

We compute the genus 0 free energy for the 2-matrix model with quartic interactions, which acts as a generating function for the Ising model's partition function on a random, 4-regular, planar graph. This is consistent with the predictions of Kazakov and Boulatov on this model, as well as subsequent confirmation of this formula using combinatorial methods. We also provide a new parametric formula for the free energy and give a characterization of the phase space. Our analysis is based on a steepest descent Riemann-Hilbert analysis of the associated biorthogonal polynomials and the corresponding isomonodromic $\tau$-function. A key ingredient in the analysis is a parametrization of the spectral curve. This analysis lays the groundwork for the subsequent study of the multicritical point, which we will study in a forthcoming work.

math-ph↗

arXiv2023

Lozenge Tilings of a Hexagon and q-Racah Ensembles

We study the limiting behavior of random lozenge tilings of the hexagon with a q-Racah weight as the size of the hexagon grows large. Based on the asymptotic behavior of the recurrence coefficients of the q-Racah polynomials, we give a new proof for the fact that that the height function for a random tiling concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian Free Field. These results were recently proved using (dynamic) loop equation techniques. In this paper we extend the recurrence coefficient approach that was developed for (dynamic) orthogonal polynomial ensembles to the setting of q-orthogonal polynomials. An interesting feature is that the complex structure is easily found from the limiting behavior of the (explicitly known) recurrence coefficients. A particular motivation for studying this model is that the variational characterization of the limiting height function has an inhomogeneous term. The study of the regularity properties of the minimizer for general variation problems with such inhomogeuous terms is a challenging open problem. We show that, in a general setup, the variational problem gives rise to a natural complex structure that is associated to the same Beltrami equation as in the homogeneous situation. We also derive a relation between the complex structure and the complex slope. In case of the q-Racah weighting of lozenge tilings of the hexagon, our representation of the limit shape and their fluctuations in terms of the recurrence coefficients allows us to verify this relation explicitly.

math.PR↗

arXiv2023

A double scaling limit for the d-PII equation with boundary conditions

We study a double scaling limit for a solution of the discrete Painlev\'e II equation with boundary conditions. The location of the right boundary point is in the critical regime where the discrete Painlev\'e equation turns into the continuous Painlev\'e II equation. Our main results it that, instead of the Hastings-McLeod solution (which would occur when the right boundary point is at infinity), the solution to the discrete equation converges in a double scaling limit to a tronqu\'ee solution of the Painlev\'e II equation that behaves like the Hastings-McLeod solution at minus infinity and has a pole at a prescribed location. Our proof of the double scaling limit is based on finding an approximation that is sufficiently close in order to apply the Kantorovich theorem for Netwons method. To meet the criteria for this theorem, we will establish a lower bound for the solutions to the Painlev\'e II equation that occur (including the Hastings-McLeod solution).

math.CA↗

arXiv2023

Benford's law and the C$\beta$E

We study the individual digits for the absolute value of the characteristic polynomial for the Circular $\beta$-Ensemble. We show that, in the large $N$ limit, the first digits obey Benford's Law and the further digits become uniformly distributed. Key to the proofs is a bound on the rate of convergence in total variation norm in the CLT for the logarithm of the absolute value of the characteristic polynomial.

math.PR↗

arXiv2022

On the domino shuffle and matrix refactorizations

This paper is motivated by computing correlations for domino tilings of the Aztec diamond. It is inspired by two of the three distinct methods that have recently been used in the simplest case of a doubly periodic weighting, that is the two-periodic Aztec diamond. One of the methods, powered by the domino shuffle, involves inverting the Kasteleyn matrix giving correlations through the local statistics formula. Another of the methods, driven by a Wiener-Hopf factorization for two-by-two matrix valued functions, involves the Eynard-Mehta theorem. For arbitrary weights the Wiener-Hopf factorization can be replaced by an LU- and UL-decomposition, based on a matrix refactorization, for the product of the transition matrices. This paper shows that, for arbitrary weightings of the Aztec diamond, the evolution of the face weights under the domino shuffle and the matrix refactorization is the same. In particular, these dynamics can be used to find the inverse of the LGV matrix in the Eynard-Mehta Theorem.

math.PR↗

arXiv2022

Biased $2 \times 2$ periodic Aztec diamond and an elliptic curve

We study a biased $2\times 2$ periodic random domino tilings of the Aztec diamond and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.

math.PR↗

arXiv2021

Global fluctuations for Multiple Orthogonal Polynomial Ensembles

We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others. Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker--Campbell--Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero. We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

math.PR↗

arXiv2019

A periodic hexagon tiling model and non-Hermitian orthogonal polynomials

We study a one-parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling. The description of the asymptotic behavior can be separated into two regimes: the low and the high temperature regime. Our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there. For low temperatures, the disorded region consists of two disjoint ellipses. In the high temperature regime the two ellipses merge into a single simply connected region. At the transition from the low to the high temperature a tacnode appears. The key to our asymptotic study is a recent approach introduced by Duits and Kuijlaars providing a double integral representation for the correlation kernel. One of the factors in the integrand is the Christoffel-Darboux kernel associated to polynomials that satisfy non-Hermitian orthogonality relations with respect to a complex-valued weight on a contour in the complex plane. We compute the asymptotic behavior of these orthogonal polynomials and the Christoffel-Darboux kernel by means of a Riemann-Hilbert analysis. After substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments.

math-ph↗

arXiv2017

The two periodic Aztec diamond and matrix valued orthogonal polynomials

We analyze domino tilings of the two-periodic Aztec diamond by means of matrix valued orthogonal polynomials that we obtain from a reformulation of the Aztec diamond as a non-intersecting path model with periodic transition matrices. In a more general framework we express the correlation kernel for the underlying determinantal point process as a double contour integral that contains the reproducing kernel of matrix valued orthogonal polynomials. We use the Riemann-Hilbert problem to simplify this formula for the case of the two-periodic Aztec diamond. In the large size limit we recover the three phases of the model known as solid, liquid and gas. We describe fine asymptotics for the gas phase and at the cusp points of the liquid-gas boundary, thereby complementing and extending results of Chhita and Johansson.

math.PR↗

arXiv2017

Relative Szegő asymptotics for Toeplitz determinants

We study the asymptotic behavior, as $n\to\infty$, of ratios of Toeplitz determinants $D_n(e^h dμ)/D_n(dμ)$ defined by a measure $μ$ on the unit circle and a sufficiently smooth function $h$. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on $h$ and only a few Verblunsky coefficients associated to $μ$. As a result, we establish a relative version of the Strong Szegő Limit Theorem for a wide class of measures $μ$ with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

math-ph↗

arXiv2017

Mesoscopic fluctuations for the thinned Circular Unitary Ensemble

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.

math.PR↗

arXiv2015

On global fluctuations for non-colliding processes

We study the global fluctuations for a class of determinantal point processes coming from large systems of non-colliding processes and non-intersecting paths. Our main assumption is that the point processes are constructed by biorthogonal families that satisfy finite term recurrence relations. The central observation of the paper is that the fluctuations of multi-time or multi-layer linear statistics can be efficiently expressed in terms of the associated recurrence matrices. As a consequence, we prove that different models which share the same asymptotic behavior of the recurrence matrices, also share the same asymptotic behavior for the global fluctuations. An important special case is when the recurrence matrices have limits along the diagonals, in which case we prove Central Limit Theorems for the linear statistics. We then show that these results prove Gaussian Free Field fluctuations for the random surfaces associated to these systems. To illustrate the results, several examples will be discussed, including non-colliding processes for which the invariant measures are the classical orthogonal polynomial ensembles and random lozenge tilings of a hexagon.

math-ph↗

arXiv2013

On mesoscopic equilibrium for linear statistics in Dyson's Brownian Motion

In this paper we study mesoscopic fluctuations for Dyson's Brownian motion with $β=2$. Dyson showed that the Gaussian Unitary Ensemble (GUE) is the invariant measure for this stochastic evolution and conjectured that, when starting from a generic configuration of initial points, the time that is needed for the GUE statistics to become dominant depends on the scale we look at: The microscopic correlations arrive at the equilibrium regime sooner than the macrosopic correlations. In this paper we investigate the transition on the intermediate, i.e. mesoscopic, scales. The time scales that we consider are such that the system is already in microscopic equilibrium (sine-universality for the local correlations), but we have not yet reached equilibrium at the macrosopic scale. We describe the transition to equilibrium on all mesoscopic scales by means of Central Limit Theorems for linear statistics with sufficiently smooth test functions. We consider two situations: deterministic initial points and randomly chosen initial points. In the random situation, we obtain a transition from the classical Central Limit Theorem for independent random variables to the one for the GUE.

math-ph↗

arXiv2013

Central Limit Theorems for Biorthogonal Ensembles and Asymptotics of Recurrence Coefficients

We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a Central Limit Theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a Central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai Class of an interval. Our results also extend previous results on Unitary Ensembles in the one-cut case. Finally, we will illustrate our results by deriving Central Limit Theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.

math-ph↗