# 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…

## 35 Citations

### ON THE ASYMPTOTIC BEHAVIOR OF A LOG GAS IN THE BULK SCALING IN THE PRESENCE A VARYING EXTERNAL POTENTIAL

- Mathematics
- 2015

. 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

- Mathematics
- 2015

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

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- 2018

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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…

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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…

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