TY - JOUR

T1 - On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential I

AU - Bothner, Thomas

AU - Deift, Percy

AU - Its, Alexander

AU - Krasovsky, Igor

N1 - Funding Information:
T. Bothner acknowledges the support of Concordia University through a postdoctoral fellow top-up award, as well as the hospitality of the Banff International Research Station, where part of this work was completed. P. Deift acknowledges support of NSF Grant DMS-1300965. A. Its acknowledges support of NSF Grants DMS-1001777 and DMS-1361856, and of SPbGU grant N 11.38.215.2014, as well as the hospitality of the Berlin Technical University and the Imperial College of London, where part of this work was completed. I.K. is grateful for the hospitality of Indiana University-Purdue University Indianapolis in April 2013 and acknowledges support of the European Community Seventh Framework grant “Random and Integrable Models in Mathematical Physics”. The authors would like to thank Jinho Baik for bringing [] to their attention. Also the authors would like to thank the referees for their careful reading of the original version of this paper, and for their comments and questions, which led the authors to reformulate, correct and streamline the paper in a variety of ways.
Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

PY - 2015/8/1

Y1 - 2015/8/1

N2 - 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-γ) supported on an interval of length $${\frac{2s}{\pi}}$$2sπ. We evaluate, in particular, the double scaling limit of $${\det(I-\gamma K_s)}$$det(I-γKs) as $${s\rightarrow\infty}$$s→∞ and $${\gamma\uparrow 1}$$γ↑1 , in the region $${0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}$$0≤κ=vs=-12sln(1-γ)≤1-δ , for any fixed $${0 < \delta < 1}$$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).

AB - 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-γ) supported on an interval of length $${\frac{2s}{\pi}}$$2sπ. We evaluate, in particular, the double scaling limit of $${\det(I-\gamma K_s)}$$det(I-γKs) as $${s\rightarrow\infty}$$s→∞ and $${\gamma\uparrow 1}$$γ↑1 , in the region $${0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}$$0≤κ=vs=-12sln(1-γ)≤1-δ , for any fixed $${0 < \delta < 1}$$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).

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U2 - 10.1007/s00220-015-2357-1

DO - 10.1007/s00220-015-2357-1

M3 - Article

AN - SCOPUS:84928724584

VL - 337

SP - 1397

EP - 1463

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -