1,720,979 research outputs found
Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results
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On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I
Full text linkWe study the determinant det(I−γKs),0<γ<1 , of the integrable Fredholm operator K s acting on the interval (−1, 1) with kernel Ks(λ,μ)=sins(λ−μ)π(λ−μ) . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature β=2 , in the presence of an external potential v=−12ln(1−γ) supported on an interval of length 2sπ . We evaluate, in particular, the double scaling limit of det(I−γKs) as s→∞ and γ↑1 , in the region 0≤κ=vs=−12sln(1−γ)≤1−δ , for any fixed 0<δ<1 . This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995)
The sine process under the influence of a varying potential
Full text linkWe review the authors’ recent work where we obtain the uniform large s asymptotics for the Fredholm determinant D(s,γ)≔det(I−γKs↾L2(−1,1)), 0 ≤ γ ≤ 1. The operator Ks acts with kernel Ks(x, y) = sin(s(x − y))/(π(x − y)), and D(s, γ) appears for instance in Dyson’s model of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned Gaussian unitary ensemble
Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities
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On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II
Full text linkIn this paper we continue our analysis [3] of the determinant det (I−γKs), γ ∈ (0,1) where K s is the trace class operator acting in L 2(−1, 1) with kernel Ks(λ,μ)=sin s(λ−μ)π(λ−μ) . In [3] various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below)
Whittaker functions and related stochastic processes
No full textWe review some recent results on connections between Brownian motion, Whittaker functions, random matrices and representation theory
Orthogonal polynomials and random matrices : a Riemann-Hilbert approach
No full textix, 261 p. : ill. ; 26 c
Orthogonal polynomials and random matrices
No full textThis volume expands on a set of lectures held at the Courant Institute on Riemann-Hilbert problems, orthogonal polynomials, and random matrix theory. The goal of the course was to prove universality for a variety of statistical quantities arising in the theory of random matrix models. The central question was the following: Why do very general ensembles of random n {\times} n matrices exhibit universal behavior as n {\rightarrow} {\infty}? The main ingredient in the proof is the steepest descent method for oscillatory Riemann-Hilbert problems
Asymptotics of polynomials orthogonal with respect to a logarithmic weight
No full textIn this talk we show how to compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight w(x)dx = [log 2k/(1-x)] dx
on (-1; 1), k > 1, and verify a conjecture of A. Magnus for these coefficients. We use
Riemann{Hilbert/steepest-descent methods, but not in the standard way as there is no
known parametrix for the Riemann{Hilbert problem in a neighborhood of the logarithmic
singularity at x = 1.
This is joint work with Oliver Conway.Non UBCUnreviewedAuthor affiliation: New York UniversityFacult
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