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Bulk Universality of General βEnsembles with Nonconvex Potential
, 2012
"... We prove the bulk universality of the βensembles with nonconvex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. ..."
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We prove the bulk universality of the βensembles with nonconvex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure ” so that the local statistics are preserved under this convexification.
Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
, 2013
"... We consideraclassicalsystem of n chargedparticlesin an external confining potential, in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n −1 (meanfield scaling). By a suitable splitting of the Hamiltonian, we extract the next ..."
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We consideraclassicalsystem of n chargedparticlesin an external confining potential, in any dimension d ≥ 2. The particles interact via pairwise repulsive Coulomb forces and the coupling parameter is of order n −1 (meanfield scaling). By a suitable splitting of the Hamiltonian, we extract the next to leading order term in the ground state energy, beyond the meanfield limit. We show that this next order term, which characterizes the fluctuations of the system, is governed by a new “renormalized energy ” functional providing a way to compute the total Coulomb energy of a jellium (i.e. an infinite set of point charges screened by a uniform neutralizing background), in any dimension. The renormalization that cuts out the infinite part of the energy is achieved by smearing out the point charges at a small scale, as in Onsager’s lemma. We obtain consequences for the statistical mechanics of the Coulomb gas: next to leading order asymptotic expansion of the free energy or partition function, characterizations of the Gibbs measures, estimates on the local charge fluctuations and factorization estimates for reduced densities. This extends results of Sandier and Serfaty to dimension higher than two by an alternative
1 BULK UNIVERSALITY FOR ONEDIMENSIONAL LOGGASES
"... In this note we consider βensembles with real analytic potential and arbitrary inverse temperature β, and review some recent universality results for these measures, obtained in joint works with L. Erdős and H.T. Yau. In the limit of a large number of particles, the local eigenvalues statistics in ..."
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In this note we consider βensembles with real analytic potential and arbitrary inverse temperature β, and review some recent universality results for these measures, obtained in joint works with L. Erdős and H.T. Yau. In the limit of a large number of particles, the local eigenvalues statistics in the bulk are universal: they coincide with the spacing statistics for the Gaussian βensembles. We also discuss the proof of the rigidity of the particles up to the optimal scale N −1+ε.
Edge Universality of Beta Ensembles
, 2013
"... We prove the edge universality of the beta ensembles for any β � 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C 4. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results ..."
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We prove the edge universality of the beta ensembles for any β � 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C 4. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class C 4.