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Igor Krasovsky

Publications and source records attributed to Igor Krasovsky.

9 recordsLinked to original sources
arXiv2022

Weak and strong confinement in the Freud random matrix ensemble and gap probabilities

The Freud ensemble of random matrices is the unitary invariant ensemble corresponding to the weight $\exp(-n |x|^{\beta})$, $\beta>0$, on the real line. We consider the local behaviour of eigenvalues near zero, which exhibits a transition in $\beta$. If $\beta\ge 1$, it is described by the standard sine process. Below the critical value $\beta=1$, it is described by a process depending on the value of $\beta$, and we determine the first two terms of the large gap probability in it. This so called weak confinement range $0<\beta<1$ corresponds to the Freud weight with the indeterminate moment problem. We also find the multiplicative constant in the asymptotic expansion of the Freud multiple integral for $\beta\ge 1$.

math-ph↗
arXiv2021

Airy-kernel determinant on two large intervals

We find the probability of two gaps of the form $(sc,sb)\cup (sa,+\infty)$, $c 0$, in the edge scaling limit of the Gaussian Unitary Ensemble of random matrices, including the multiplicative constant in the asymptotics.

math.FA↗
arXiv2020

Sine-kernel determinant on two large intervals

We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.

math-ph↗
arXiv2019

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum

We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matrices and obtain a representation of the critical almost Mathieu family that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than 1/2 for all irrational frequencies, solving a long-standing problem. Other corollaries include two very elementary proofs of zero measure of the spectrum (Problem 5 in [41]) and a similar Hausdorff dimension result for the quantum graph graphene.

math.SP↗
arXiv2018

The sine process under the influence of a varying potential

We review the authors' recent work \cite{BDIK1,BDIK2,BDIK3} where we obtain the uniform large $s$ asymptotics for the Fredholm determinant $D(s,\gamma):=\det(I-\gamma K_s\upharpoonright_{L^2(-1,1)})$, $0\leq\gamma\leq 1$. The operator $K_s$ acts with kernel $K_s(x,y)=\sin(s(x-y))/(\pi(x-y))$ and $D(s,\gamma)$ appears for instance in Dyson's model \cite{Dyson2} of a Coulomb log-gas with varying external potential or in the bulk scaling analysis of the thinned GUE \cite{BP}.

math-ph↗
arXiv2017

Splitting of a gap in the bulk of the spectrum of random matrices

We 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.

math.FA↗
arXiv2015

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II

In this paper we continue our analysis \cite{BDIK} of the determinant $\det(I-\gamma K_s),\gamma\in(0,1)$ where $K_s$ is the trace class operator acting in $L^2(-1,1)$ with kernel $K_s(\lambda,\mu)=\frac{\sin s(\lambda-\mu)}{\pi(\lambda-\mu)}$. In \cite{BDIK} 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).

math-ph↗
arXiv2014

On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I

We study the determinant $\det(I-\gamma K_s), 0<\gamma <1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $\beta=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-\gamma)$ supported on an interval of length $\frac{2s}{\pi}$. We evaluate, in particular, the double scaling limit of $\det(I-\gamma K_s)$ as $s\rightarrow\infty$ and $\gamma\uparrow 1$, in the region $0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta$, for any fixed $0<\delta<1$. This problem was first considered by Dyson in \cite{Dy1}.

math-ph↗