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F2018 N 6 9 iyj 15:00 -- 17:30
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1. 15:00--16:00
u: Khanh Duy Trinh ikEj

TitleFClassical beta ensembles in global regime

Abstract: Gaussian beta ensembles, Wishart beta ensembles and Jacobi beta ensembles are three typical examples of beta ensembles on the real line. The limiting behavior of the empirical distribution, the probability measure putting equal mass at each eigenvalue, is one of fundamental problem in studying a random matrix model. Based on their random tridiagonal matrix models, this talk introduces a unify approach to completely solve that problem.The result can be shortly mentioned as follows. The limit of the empirical distribution depends only on the limit of the coupling $$n \beta$$: zero, finite or infinite. Here $$n$$ is the system size, or the order of a random matrix, and $$n \beta$$ is a parameter regarded as the inverse temperature of the system. In particular, the case where $$n \beta$$ tends to infinity is similar to the case of fixed beta. When $$n \beta$$ tends to a finite constant, the limiting distribution is an associated version of the corresponding weight in the sense of orthogonal polynomials.

16:00--16:30 tea time

2. 16:30--17:30
u: ؁@ isElԊj

Title: Asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field

AbstractFAsymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi in terms of the Fredholm determinant. In the subclass, the obtained leading orders are same with the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point field associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of the states are also investigated in the case where the single site potentials have compact supports.

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