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Quenched LDP for words in a letter sequence | | Matthias Birkner
; Andreas Greven
; Frank den Hollander
; | | Date: |
16 Jul 2008 | | 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 ^d, d geq 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. | | Source: | arXiv, 0807.2611 | | Services: | Forum | Review | PDF | Favorites | |
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