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SUMMARY:Random matrices: Universality of ESDs and the circular law - Van V
 u (Rutgers University)
DTSTART:20090205T140000Z
DTEND:20090205T150000Z
UID:TALK16803@talks.cam.ac.uk
CONTACT:Andrew Thomason
DESCRIPTION:Given an n by n complex matrix $A$\, let \\mu(A) be the empiri
 cal spectral distribution (ESD) of its eigenvalues $\\lambda_i\, i=1\, ...
 \, n$.\n\nWe consider the limiting distribution of the normalized ESD $\\m
 u_{\\frac{1}{\\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})_{1 \\leq 
 i\,j \\leq n}$ where the random variables $a_{ij} - \\E(a_{ij})$ are iid c
 opies of a fixed random variable $x$ with unit variance.  We prove the ``u
 niversality principle"\, namely that the limit distribution in question is
  independent of the actual choice of $x$. In particular\, in order to comp
 ute this distribution\, one can assume that $x$ is real of complex\ngaussi
 an.\n\nAs a corollary we establish the Circular Law conjecture in full gen
 erality. The proof combines ideas from several areas of mathematics: addit
 ive combinatorics\, theoretical computer science\, probability and high\nd
 imensional geometry.\n\nJoint work with Terence Tao.\n\n
LOCATION:MR12
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