Abstract
We give a sufficient criterion for the weak disorder regime of directed polymers in random environment, which extends a well-known second moment criterion. We use a stochastic representation of the size-biased law of the partition function.
Highlights
We think of the graph of Sn as the polymer, which is influenced by the random environment generated by the ξ(x, n) through a reweighting of paths with en := en(ξ, S) := exp n j=1 βξ(Sj, j) − λ(β), that is, we are interested in the random probability measures on path space given by μn(ds)
Our aim here is to give a condition for the weak disorder regime
Proposition 1 If λ(2β) − 2λ(β) < log α∗, limn→∞ Zn > 0 almost surely, that is, the directed polymer is in the weak disorder regime
Summary
We think of the graph of Sn as the (directed) polymer, which is influenced by the random environment generated by the ξ(x, n) through a reweighting of paths with en := en(ξ, S) := exp n j=1 βξ(Sj, j) − λ(β) , that is, we are interested in the random probability measures on path space given by μn(ds). We denote their cumulant generating function by λ(β) := log E[exp(βξ(x, n))]. Proposition 1 If λ(2β) − 2λ(β) < log α∗, limn→∞ Zn > 0 almost surely, that is, the directed polymer is in the weak disorder regime.
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