A Fuchsian Matrix Differential Equation for Selberg Correlation Integrals

作者：Forrester Peter J^*; Rains Eric M

来源：Communications in Mathematical Physics, 2012, 309(3): 771-792.

DOI：10.1007/s00220-011-1305-y

We characterize averages of Pi(N)(l=1) vertical bar x -t(l)|(alpha-1) with respect to the Selberg density, further constrained so that t(l) is an element of [0, x](l = 1,..., q) and t(l) is an element of [x, 1](l = q + 1,..., N), in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated, thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able to compute power series for the marginal distributions of the t(j) (j = 1,..., N). In the case q = 0 and alpha < 1 we compute the explicit leading order term in the x -> 0 asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations. In the case q = 0 and alpha is an element of Z(+), and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured.

• 出版日期2012-2