Almost-Sure Central Limit Theorem for Directed Polymers and Random Corrections |
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Authors: | C Boldrighini RA Minlos A Pellegrinotti |
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Institution: | (1) {Dipartimento di Matematica e Fisica, Universita di Camerino, via Madonna delle Carceri 9, 62032 Camerino, Italy, IT;(2) Institute for Problems of Information Transmission, Russian Academy of Sciences, RU;(3) Dipartimento di Matematica, Universita degli studi de Roma Tre, Via C. Segre 2, 00146 Roma, Italy, IT |
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Abstract: | We consider a general model of directed polymers on the lattice , weakly coupled to a random environment. We prove that the central limit theorem holds almost surely for the discrete time
random walk X
T
associated to the polymer. Moreover we show that the random corrections to the cumulants of X
T
are finite, starting from some dimension depending on the index of the cumulants, and that there are corresponding random
corrections of order , , in the asymptotic expansion of the expectations of smooth functions of X
T
. Full proofs are carried out for the first two cumulants. We finally prove a kind of local theorem
showing that the ratio of the probabilities of the events to the corresponding probabilities with no randomness, in the region of “moderate” deviations from the average drift bT, are, for almost all choices of the environment, uniformly close, as , to a functional of the environment “as seen from (T,y)$”.
Received: 14 October 1996 / Accepted: 28 March 1997 |
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Keywords: | |
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