The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature beta = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions R-d and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions C-d, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For C-d with d = 1, this corresponds to non-interacting fermions in a rotating trap.

We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in [20]. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.

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.

We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function 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.

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter epsilon > 0. The domain has two cusps, one pointing up and one pointing down. In the limit epsilon down arrow 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime epsilon down arrow 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.

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.

We consider an ensemble of N discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as N -> infinity. In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.

We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers under fairly weak conditions on the underlying measure mu. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.

We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

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 disordered 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. 

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.

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize recent results by Kuijlaars and one of the authors [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 173-196] that were concerned with banded Toeplitz matrices.

We study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two-dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field.

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 that 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.

After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4x4 Riemann-Hilbert problem related to the Painlev, II equation. Also an open problem will be presented.

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 Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials. (C) 2021 The Authors. Published by Elsevier Inc.

We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4×4 Riemann-Hilbert problem related to the Hastings-McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.

We consider the two sequences of biorthogonal polynomials (p(k,n))(k=0)(infinity) and (q(k,n))(k=0)(infinity) related to the Hermitian two-matrix model with potentials V (x) = x(2)/2 and W(y) = y(4)/4 + ty(2). From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p(n,n) as n -> infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t.

We study the asymptotic behaviour, as n -> infinity, of ratios of Toeplitz determinants D-n(e(h)d mu)/D-n(d mu) defined by a measure mu 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 mu. As a result, we establish a relative version of the Strong Szego Limit Theorem for a wide class of measures mu with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

We consider the two matrix model with an even quartic potential W(y) = y 4/4+ ay 2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1· Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

We study the limiting eigenvalue distribution of n x n banded Toeplitz matrices as n -> infinity. From classical results of Schmidt, Spitzer, and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n -> infinity. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimizes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.

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 the 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.

The eigenvalue statistics of a pair (M(1), M(2)) of n x n Hermitian matrices taken randomly with respect to the measure 1/Z(n) exp (-n Tr(V(M(1)) + W(M(2)) - tau M(1)M(2)))dM(1) dM(2) can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 x 4 matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y(4)/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M(1) (when averaged over M(2)) in the global and local regime as n -> infinity in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.