17 results found
Jitomirskaya S, Konstantinov L, Krasovsky I, 2020, On the spectrum of critical almost Mathieu operators in the rational case, Journal of Spectral Theory, ISSN: 1664-039X
We derive a new Chambers-type formula and prove sharper upper bounds on the measure of the spectrum of critical almost Mathieu operators with rational frequencies.
Fahs B, Krasovsky I, 2019, Splitting of a gap in the bulk of the spectrum of random matrices, Duke Mathematical Journal, Vol: 168, Pages: 3529-3590, ISSN: 0012-7094
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.
Krasovsky I, Bothner T, Its A, et al., 2018, The sine process under the influence of a varying potential, Journal of Mathematical Physics, ISSN: 1089-7658
Krasovsky I, 2017, Central Spectral Gaps of the Almost Mathieu Operator, Communications in Mathematical Physics, Vol: 351, Pages: 419-439, ISSN: 0010-3616
We consider the spectrum of the almost Mathieu operator Hα with frequency αand in the case of the critical coupling. Let an irrational α be such that |α − pn/qn| < cq−κn,where pn/qn, n = 1, 2, . . . are the convergents to α, and c, κ are positive absolute constants,κ < 56. Assuming certain conditions on the parity of the coefficients of the continuedfraction of α, we show that the central gaps of Hpn/qn, n = 1, 2, . . . , are inherited as spectralgaps of Hα of length at least c0q−κ/2n , c0 > 0.
Claeys T, Krasovsky I, 2015, Toeplitz determinants with merging singularities, Duke Mathematical Journal, Vol: 164, Pages: 2897-2987, ISSN: 0012-7094
We study asymptotic behavior for the determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance 2t≥0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0<t<t0, where t0 is fixed. They describe the transition as t→0 between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.
Bothner T, Deift P, Its A, et al., 2015, On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I, Communications in Mathematical Physics, Vol: 337, Pages: 1397-1463, ISSN: 0010-3616
We 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).
Deift P, Its A, Krasovsky I, 2013, Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results, COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, Vol: 66, Pages: 1360-1438, ISSN: 0010-3640
Krasovsky I, Woess W, 2011, Aspects of Toeplitz Determinants, Alp Workshop, Publisher: BIRKHAUSER VERLAG AG, Pages: 305-324, ISSN: 1050-6977
Deift P, Its A, Krasovsky I, 2011, Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants withFisher-Hartwig singularities, Annals of Mathematics, Vol: 174, Pages: 1243-1299
Claeys T, Its A, Krasovsky I, 2011, Emergence of a singularity for Toeplitz determinants and Painlev\'e V, Duke Mathematical Journal, Vol: 160, Pages: 207-262
Deift P, Krasovsky I, Vasilevska J, 2010, Asymptotics for a determinant with a confluent hypergeometric kernel, International Mathematics Research Notices
Claeys T, Its A, Krasovsky I, 2010, Higher order analogues of the Tracy-Widom distribution and the Painlev\'e II hierarchy, Commun Pure Appl Math, Vol: 63, Pages: 362-412, ISSN: 0010-3640
Krasovsky I, 2009, Large Gap Asymptotics for Random Matrices, 15th International Congress on Mathematical Physics, Publisher: SPRINGER-VERLAG BERLIN, Pages: 413-419
Deift P, Its A, Krasovsky I, 2008, Asymptotics of the Airy-Kernel determinant, COMMUNICATIONS IN MATHEMATICAL PHYSICS, Vol: 278, Pages: 643-678, ISSN: 0010-3616
Krasovsky IV, 2007, Correlations of the characteristic polynomials in the gaussian unitary ensemble or a singular hankel determinant, DUKE MATHEMATICAL JOURNAL, Vol: 139, Pages: 581-619, ISSN: 0012-7094
Deift P, Its A, Krasovsky I, et al., 2007, The Widom-Dyson constant for the gap probability in random matrix theory, Joint Meeting on Integrable Systems and Special Functions/Integrable Systems, Orthogonal Polynomials, and Random Matrices - Interdisciplinary Aspects, Publisher: ELSEVIER SCIENCE BV, Pages: 26-47, ISSN: 0377-0427
Deift P, Its A, Krasovsky I, Eigenvalues of Toeplitz matrices in the bulk of the spectrum, Bull. Inst. Math. Acad. Sin. (N. S.) 7 (2012), 437-461
The authors analyze the asymptotics of eigenvalues of Toeplitz matrices withcertain continuous and discontinuous symbols. In particular, the authors provea conjecture of Levitin and Shargorodsky on the near-periodicity of Toeplitzeigenvalues.
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