Download Communications In Mathematical Physics - Volume 278 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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We will show that the leading order coefficient, e0 (x, t), of the asymptotic partition (k) (k) function, k −2 log(Z N (t)/Z N (0)) is found in terms of the equilibrium measure µ = µVt /x . This leading order coefficient, which satisfies a potential-theoretic variational problem, has been studied from several perspectives in the literature. We point out that what is novel here is that we identify a general expression for this coefficient which has a similar form for all external fields of the form (1/2)λ2 + t2ν λ2ν .

Remark. 3) is a fundamental component in the theory of random matrices, as well as integrable systems and approximation theory. It is well known, (see, for example, [29]), that under general assumptions on V , the infimum is achieved at a unique measure µV , called the equilibrium measure. For external fields V that are analytic in a neighborhood of the real axis, and with sufficient growth at ∞, the equilibrium measure is supported on finitely many intervals, with density that is analytic on the interior of each interval, behaving at worst like a square root at each endpoint, (see [10] and [11]).

This ODE is solved implicitly by a solution to the algebraic equation 1 = z 0 (s) − cν sz 0 (s)ν . 1 (5), this relation can also be derived directly from the characterization of the equilibrium measure. 3) allows us to write s as well as derivatives of z 0 as rational functions of z 0 . This will be exploited to arrive at the explicit representations of the z g given below. 40 N. M. Ercolani et al. 5) (ν) and F [2r ] denotes the coefficient of k −2r in F . We note that the terms in Forcingg |w=1 depend only on z j , j < g and their derivatives.

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