Optimal tail estimates for directed last passage site percolation with geometric random variables

Abstract

e-print archive: http://xxx.lanl.gov/math.PR/0112162 1208 BAIK, DEIFT, MCLAUGHLIN, MILLER, AND ZHOUIn this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.

Equality of the Spectral and Dynamical Definitions of Reflection

“…One-sided CMV matrices appeared first in the numeric matrix literature [1,2,33] and were rediscovered by the OPUC community [5]. Two-sided CMV matrices were defined first in [40], although related objects appeared earlier in [3,10].…”

“…Similarly, we can change the summation limits of the k sums in t (1) , t (2) to any other finite value, since in the limit, finite sums multiplied by Im z go to zero. The result is…”

Quadratic Hermite–Padé Approximation to the Exponential Function: A Riemann–Hilbert Approach

“…The other two curves are unbounded and lie in the right half-plane. They join z 3 and z 4 with infinity and we call them Γ E, 3 and Γ E,4 , respectively.…”

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“…The steepest descent method for an asymptotic solution of this problem was suggested by Deift and Zhou [18]. The matrix Riemann-Hilbert problem method has not only made it possible to make considerable progress in the asymptotic theory of polynomials orthogonal with respect to a non-constant real-valued weight function (see [3,4,19,20,21]), but also proved capable of being extended to the case of complex-valued non-constant weights in the complex plane [2,6,9]. Apart from the above references, a detailed description of this method is contained in the monograph [22]; an approachable exposition of the main ideas can also be found in [23].…”

“…In [1,2], they were used to determine the rate of approximation of analytic functions by rational functions. Among many other impressive applications we note universality theorems in the theory of random matrices [3,4,5], applications to asymptotic combinatorics [6,7] and mathematical physics [8,9].…”