The Algebra & Geometry of Modern Physics

The scalar products and certain correlation functions of models solvableby the quantum separation of variables can be expressed in terms of$N$-fold multiple integrals which can be thought of as the partitionfunction of a one dimensionalgas of particles trapped in an externalpotential $V$ and interacting through repulsive two-body interactions ofthe type $\ln \big[ \sinh[\pi \omega_1(\lambda-\mu)]\cdot \sinh[\pi \omega_2(\lambda-\mu)] \big]$. The analysis of the large-$N$ asymptotic behaviour ofthese integrals is of interest to the description of the continuum limitof the integrable model. Although such partition functions present certainstructural resemblances with those arising in the context of the so-called$\beta$-ensembles, their large-$N$ asymptotic analysis demands theintroduction of several new ingredients. Such a complication in theanalysis is due to the lack of dilation invariance of the exponential ofthe two-body interaction. In this talk, I shall discuss the main featuresof the method of asymptotic analysis which we have developed. The methodutilises large-deviation techniques on the one hand and theRiemann--Hilbert problem approach to truncated Wiener-Hopfsingular-integral equations on the other hand. This is a joint work withG. Borot (Max-Planck Institut, Bonn, Germany) and A. Guionnet (MIT,Boston, USA).