On the Large Deviations Rate Function for Symmetric Simple Random Walk in Dimension d ∈ Z

The large deviations rate function Lambda((d))of d-dimensional symmetric simple random walk S-(d)(n) describes the log-asymptotic behavior of the fundamental probability P(S-(d)(n) =x)in the large deviations regime, that is, when n=O parallel to x parallel to, but also in the so-called moderate deviations regime, that is, when n 1/2=o(parallel to x parallel to) and parallel to x parallel to=o(n). In Benes (2019), we provided precise asymptotics forP(S(d)(n) =x)in dimensions 1 and 2 and, as an immediate consequence, we obtained Lambda(d) for d= 1 and d= 2. The techniques developed in that paper do not translate readily to higher dimensions. In the present paper, we show that for d is an element of Z+, Lambda((d))(alpha) = logd+12dXi=1(ai+alpha i) log(ai+alpha i) + (ai-alpha i) log(ai-alpha i), where alpha= (alpha(1),...,alpha(d)) satisfies Sigma(d)i=1|alpha(i)| <= 1. The quantities a(i)=ai(alpha) are not explicit but are given in terms of the solution of a system of quadratic equations. This result represents partialprogress towards the generalization of the results in Bene & scaron; (2019) to any dimension d is an element of Z(+).