Complex random energy model: zeros and fluctuations

Complex random energy model : zeros and fluctuations

The partition function of the random energy model at inverse temperature β is defined by ZN (β) = ∑N k=1 exp(β √ nXk), where X1, X2, . . . are independent real standard normal random variables, and n = logN . We identify the asymptotic structure of complex zeros of ZN , as N → ∞, confirming predictions made in the theoretical physics literature. Also, we describe the limiting complex fluctuatio...

The Complex Zeros of Random Polynomials

Mark Kac gave an explicit formula for the expectation of the number, vn (a), of zeros of a random polynomial, n-I Pn(z) = E ?tj, j=O in any measurablc subset Q of the reals. Here, ... ?In-I are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which E vn(L) = Jgn(x) dx. Here, we extend this formula to obtain an explicit form...

Convergence of Random Zeros on Complex Manifolds

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N , orthonormalized on a regular compact set K ⊂ Cm, almost surely converge to the equilibrium measure on K as N → ∞.

The Complex Zeros of Random Sums

Mark Kac gave an explicit formula for the expectation of the number, νn(Ω), of zeros of a random polynomial, Pn(z) = n ∑ j=0 ηjz j , in any measurable subset Ω of the reals. Here, η0, . . . , ηn are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function gn for which Eνn(Ω) = ∫ Ω gn(x)dx. Inspired by that result, Larry Shepp and I found ...

Sample to sample fluctuations in the random energy model