1 2 Fe b 20 02 Products of Random Matrices

- th / 0 20 20 56 v 1 1 9 Fe b 20 02 Comment on “ Spectroscopy with Random and Displaced Random Ensemble ”

ar X iv : m at h / 07 02 36 0 v 1 [ m at h . N A ] 1 3 Fe b 20 07 On the number of minima of a random polynomial ∗

We give an upper bound in O(d) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d)Pn, where Pn is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hy...

The abelian sandpile model on a random binary tree

We study the abelian sandpile model on a random binary tree. Using a transfer matrix approach introduced by Dhar & Majumdar, we prove exponential decay of correlations, and in a small supercritical region exponential decay of avalanche sizes. This shows a phase transition phenomenon between exponential decay and power law decay of avalanche sizes. Our main technical tools are: (1) A recursion f...

ar X iv : 0 70 8 . 28 95 v 4 [ m at h . PR ] 2 1 Fe b 20 08 RANDOM MATRICES : THE CIRCULAR LAW

Let x be a complex random variable with mean zero and bounded variance σ 2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ 1 ,. .. , λn be the eigenvalues of 1 σ √ n Nn. Define the empirical spectral distribution µn of Nn by the formula µn(s, t) := 1 n #{k ≤ n|Re(λ k) ≤ s; Im(λ k) ≤ t}. The following well-known conjecture has been open since the 1950's: Circul...

ar X iv : 0 70 8 . 28 95 v 5 [ m at h . PR ] 2 9 Fe b 20 08 RANDOM MATRICES : THE CIRCULAR LAW

Let x be a complex random variable with mean zero and bounded variance σ 2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ 1 ,. .. , λn be the eigenvalues of 1 σ √ n Nn. Define the empirical spectral distribution µn of Nn by the formula µn(s, t) := 1 n #{k ≤ n|Re(λ k) ≤ s; Im(λ k) ≤ t}. The following well-known conjecture has been open since the 1950's: Circul...