Status  Confirmed 
Seminar Series  SEMLPTHE 
Subjects  condmat,condmat.statmech,hepth,math 
Date  Friday 18 May 2018 
Time  11:00 
Institute  LPTHE 
Seminar Room  Bibliothèque 
Speaker's Last Name  Schehr 
Speaker's First Name  Gregory 
Speaker's Email Address  
Speaker's Institution  LPTMS Orsay 
Title  Noninteracting trapped fermions: from GUE to multicritical matrix models 
Abstract  We will discuss a system of N onedimensional free fermions confined by a harmonic well. At zero temperature,, this system is intimately connected to random matrices belonging to the Gaussian Unitary Ensemble (GUE). In particular, the spatial density of fermions has, for large N, a finite support and it is given by the Wigner semicircular law. Besides, close to the edges of the support, the spatial quantum fluctuations are described by the socalled AiryKernel, which plays an important role in random matrix theory. We will then focus on the joint statistics of the momenta, with a particular focus on the largest one $p_{\rm max}$. For the harmonic trap, momenta and positions play a symmetric role and hence the joint statistics of momenta is identical to that of the positions. Here we show that novel ``momentum edge statistics'' emerge when the curvature of the potential vanishes, i.e. for "flat traps" near their minimum, with $V(x) \sim x^{2n}$ and $n>1$. These are based on generalisations of the Airy kernel that we obtain explicitly. The fluctuations of $p_{\rm max}$ are governed by new universal distributions determined from the $n$th member of the second Painlevé hierarchy of nonlinear differential equations, with connections to multicritical random matrix models, which have been discussed, in the past, in the string theory literature. 
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