1,720,973 research outputs found
Splitting of a gap in the bulk of the spectrum of random matrices
Full text linkWe consider the probability of having two intervals (gaps) without eigenvalues in the bulk scaling limit of the Gaussian unitary ensemble of random matrices. We describe uniform asymptotics for the transition between a single large gap and two large gaps. For the initial stage of the transition, we explicitly determine all the asymptotic terms (up to the decreasing ones) of the logarithm of the probability. We obtain our results by analyzing double-scaling asymptotics of a Toeplitz determinant whose symbol is supported on two arcs of the unit circle
Airy-kernel determinant on two large intervals
Full text linkWe find the probability of two gaps of the form (sc,sb)∪(sa,+∞), c<0, for large s>0, in the edge scaling limit of the Gaussian Unitary Ensemble, including the multiplicative constant in the asymptotics
Aspects of Toeplitz Determinants
No full textWe review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szego, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painlevé V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results
Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results
Full text linkOn 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
Higher-Order Analogues of the Tracy-Widom Distribution and the Painleve II Hierarchy
Full text linkWe study Fredholm determinants related to a family of kernels that describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher-order analogues of the Airy kernel and are built out of functions associated with the Painleve I hierarchy. The Fredholm determinants related to those kernels are higher-order generalizations of the Tracy-Widom distribution. We give an explicit expression for the determinants in terms of a distinguished smooth solution to the Painleve II hierarchy. In addition, we compute large gap asymptotics for the Fredholm determinants. (c) 2009 Wiley Periodicals, Inc
Some problems in the spectral theory of the almost Mathieu operator
Full text linkI study the spectral properties of the almost Mathieu operator (AMO)—a well-known and well-studied self-adjoint bounded linear operator on l^2(Z) that is also a one-dimensional discrete Schrodinger operator. To begin with, by exploiting the chiral gauge transformation of the AMO, I derive a new formula in the style of the celebrated Chambers formula. This result, together with a perturbation theory approach based on Lidskii’s inequalities, is then used to obtain a sharper upper bound for the Lebesgue measure of the spectrum of the AMO in the case of rational frequency and critical coupling. I continue by establishing several estimates for the sizes of spectral gaps and bands, again for the case of rational frequency, but this time specifically for frequency having odd denominator. This latter work is conducted for a general (i.e., possibly non-critical) coupling constant, thereby generalising some key critical-case results of Krasovsky. The dissertation concludes with an appendix containing a rigorous proof for a variational principle due to Weyl that is occasionally quoted without reference in literature on the AMO.Open Acces
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)
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