On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I

@article{Bothner2014OnTA,
  title={On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I},
  author={Thomas Bothner and Percy Deift and Alexander Its and I. V. Krasovsky},
  journal={Communications in Mathematical Physics},
  year={2014},
  volume={337},
  pages={1397-1463}
}
We study the determinant $${\det(I-\gamma K_s), 0 < \gamma < 1}$$det(I-γKs),0<γ<1 , of the integrable Fredholm operator Ks acting on the interval (−1, 1) with kernel $${K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}$$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 $${\beta=2}$$β=2 , in the presence of an external potential $${v=-\frac{1}{2}\ln(1-\gamma)}$$v=-12ln(1… 
ON THE ASYMPTOTIC BEHAVIOR OF A LOG GAS IN THE BULK SCALING IN THE PRESENCE A VARYING EXTERNAL POTENTIAL
. We study the determinant det( I − γK s ) , 0 < γ < 1, of the integrable Fredholm operator K s acting on the interval ( − 1 , 1) with kernel K s ( λ,µ ) = sin s ( λ − µ ) π ( λ − µ ) . This
Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I. Communications in Mathematical Physics, 337, 1397-1463. https://doi.org/10.1007/s00220-015-2357-1
We study the determinant det(I − γKs), 0 < γ < 1, of the integrable Fredholm operator Ks acting on the interval (−1, 1) with kernel Ks(λ, μ) = sin s(λ−μ) π(λ−μ) . This determinant arises in the
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
Transition asymptotics for the Painlevé II transcendent
We consider real-valued solutions $u=u(x|s),x\in\mathbb{R}$ of the second Painlev\'e equation $u_{xx}=xu+2u^3$ which are parametrized in terms of the monodromy data
Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics
We analyze the left-tail asymptotics of deformed Tracy–Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft
On the deformed Pearcey determinant
Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for step-like initial data
We consider the compressive wave for the modified Korteweg–de Vries equation with background constants $$c>0$$c>0 for $$x\rightarrow -\infty $$x→-∞ and 0 for $$x\rightarrow +\infty $$x→+∞. We study
From gap probabilities in random matrix theory to eigenvalue expansions
We present a method to derive asymptotics of eigenvalues for trace-class integral operators K : L 2 ( J ; d λ ) ⥀ ?> , acting on a single interval J ⊂ R ?> , which belongs to the ring of integrable
Incomplete Determinantal Processes: From Random Matrix to Poisson Statistics
  • G. Lambert
  • Mathematics
    Journal of Statistical Physics
  • 2019
We study linear statistics of a class of determinantal processes which interpolate between Poisson and GUE/Ginibre statistics in dimension 1 or 2. These processes are obtained by performing an
Entanglement entropies of an interval in the free Schrödinger field theory at finite density
Abstract We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a
...
...

References

SHOWING 1-10 OF 42 REFERENCES
The Widom-Dyson constant for the gap probability in random matrix theory
Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Ferroelectric Phase
This is a continuation of the paper [4] of Bleher and Fokin, in which the large n asymptotics is obtained for the partition function Zn of the six-vertex model with domain wall boundary conditions in
Exact Solution of the Six‐Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase
We obtain the large‐n asymptotics of the partition function Zn of the six‐vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a = sinh(γ − t), b
Toeplitz determinants with merging singularities
We study asymptotic behavior for 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
UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with
Emergence of a singularity for Toeplitz determinants and Painlevé V
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
Dyson's Constant in the Asymptotics of the Fredholm Determinant of the Sine Kernel
AbstractWe prove that the asymptotics of the Fredholm determinant of I−Kα, where Kα is the integral operator with the sine kernel on the interval [0, α], are given by This formula was conjectured
Differential Equations for Quantum Correlation Functions
The quantum nonlinear Schrodinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation
...
...