Quenched LDP for words in a letter sequence

M. Birkner, A. Greven, W.Th.F. Hollander, den

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Abstract

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. We apply our LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d = 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments.
Original languageEnglish
Place of PublicationEindhoven
PublisherEurandom
Number of pages56
Publication statusPublished - 2008

Publication series

NameReport Eurandom
Volume2008034
ISSN (Print)1389-2355

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