Tomorrow's seminar

[Cond.Matt. & Stat.Mech.Section]

JOINT ICTP/SISSA STATISTICAL PHYSICS SEMINAR



Tuesday, 20 April   -   11:00 hrs.



Luigi Stasi Seminar Room- ICTP Leonardo Building - 1st floor



Pierpaolo VIVO      (I.C.T.P.)



"How many eigenvalues of a   Gaussian matrix are positive?"



Abstract



The index of a random matrix (i.e. the number of positive or negative  
eigenvalues) is a random variable providing information about the  
stability of stationary points in high-dimensional potential  
landscapes.  For a Gaussian matrix model of large size N, typically  
half of the eigenvalues are positive and half negative (Wigner's  
semicircle law), however atypical fluctuations around the semicircle  
are quite interesting and surprisingly not well understood.  Using a  
Coulomb gas technique and functional methods, we find that the  
distribution of the index is not strictly Gaussian around the mean due  
to an unusual logarithmic singularity in the rate function.  The  
variance of the index increases logarithmically with the matrix size,  
and this finding is confirmed via a comparison with an exact finite N  
result based on the Andrejeff integration formula.  The combinatorics  
behind such formula is also elucidated.



Reference: -Phys. Rev. Lett. 103, 220603 (2009)