Emergence of a singularity for Toeplitz determinants and Painlevé V

Abstract

We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter t. For t positive, the symbols are regular so that the determinants obey Szegő's strong limit theorem. If t = 0, the symbol possesses a Fisher-Hartwig singularity. Letting t → 0 we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.Using the RH properties of S, N , and P , we obtain the following. RH problem for R(a) R is analytic in C \ Σ R , where Σ R is the union of ∂U and the parts of Σ 1 , Σ 2 lying outside U (see Figure 4). (3.29) and J k (z), k = 1, 2 are the jump matrices (3.3), (3.5) of S. (c) As z → ∞, R(z) = I + O(z −1 ). Using (3.3), (3.5), and (3.23), we observe a crucial fact: J R (z) = I + O(n −1 ), for z ∈ ∂U , (3.30) J R (z) = I + O(e −cn ), for z ∈ Σ R \ ∂U , c > 0 (3.31)

Random Matrices with Merging Singularities and the Painlevé V Equation

“…The first one, σ − α (s), is analytic and real for s ∈ (0, +∞), and it has the asymptotics where Γ is Euler's Γ-function. The existence of such a solution was proved in [4]. The second solution σ + α (s) is analytic and such that σ + α (s) + Its existence was proved in [5].…”

“…The second solution σ + α (s) is analytic and such that σ + α (s) + Its existence was proved in [5]. It should be noted that uniqueness of the solutions σ ± α satisfying the above asymptotic conditions was not proven in [4,5]. Those Painlevé solutions σ − α and σ + α can be characterized in terms of Riemann-Hilbert (RH) problems; we will do this in Section 2 for σ − α and in Section 3 for σ + α .…”

“…Our proofs of Theorems 1.1 and 1.7 are based on an asymptotic analysis of the orthogonal polynomials on the real line defined by (1.19), using RH methods [7,11,12]. The asymptotic analysis for those orthogonal polynomials shows similarities to the one performed in [4,5] (following the general method from [8]) for orthogonal polynomials on the unit circle with respect to weights with merging or emerging singularities, where they were used to obtain asymptotics for Toeplitz determinants. Nevertheless, there is no known way to directly derive the asymptotics for the Hankel determinant (1.5), with respect to the n-dependent weight w n supported on the full real line, from the asymptotics for Toeplitz determinants.…”

“…For our purposes, the parameter β from [4] is set to zero. The RH problem depends on parameters α > −1/2 and s ∈ C. The relevant case for us will be s > 0.…”

“…We recall from [4] (in particular Theorem 1.8(ii) and Section 4.3 in that paper) that the RH solution Ψ = Ψ − solves a system of linear differential equations 6) where the matrices A and B have the form…”