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.

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.

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 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 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.

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. 

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.

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 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.