8,777 research outputs found
Circular law for random discrete matrices of given row sum
Let be a random matrix of size and let
be the eigenvalues of . The empirical spectral
distribution of is defined as \mu_{M_n}(s,t)=\frac{1}{n}#
\{k\le n, \Re(\lambda_k)\le s; \Im(\lambda_k)\le t\}.
The circular law theorem in random matrix theory asserts that if the entries
of are i.i.d. copies of a random variable with mean zero and variance
, then the empirical spectral distribution of the normalized matrix
of converges almost surely to the uniform
distribution \mu_\cir over the unit disk as tends to infinity.
In this paper we show that the empirical spectral distribution of the
normalized matrix of , a random matrix whose rows are independent random
vectors of given row-sum with some fixed integer satisfying
, also obeys the circular law. The key ingredient is a new
polynomial estimate on the least singular value of
Non-abelian Littlewood-Offord inequalities
In 1943, Littlewood and Offord proved the first anti-concentration result for
sums of independent random variables. Their result has since then been
strengthened and generalized by generations of researchers, with applications
in several areas of mathematics.
In this paper, we present the first non-abelian analogue of Littlewood-Offord
result, a sharp anti-concentration inequality for products of independent
random variables.Comment: 14 pages Second version. Dependence of the upper bound on the matrix
size in the main results has been remove
Central limit theorems for Gaussian polytopes
Choose random, independent points in according to the standard
normal distribution. Their convex hull is the {\sl Gaussian random
polytope}. We prove that the volume and the number of faces of satisfy
the central limit theorem, settling a well known conjecture in the field.Comment: to appear in Annals of Probabilit
Random matrices: Law of the determinant
Let be an by random matrix whose entries are independent real
random variables with mean zero, variance one and with subexponential tail. We
show that the logarithm of satisfies a central limit theorem. More
precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf
{P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log
n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le
x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Elementary proofs of Berndt's reciprocity laws
Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz
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