A condition for weak disorder for directed polymers in random environment

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. We consider the so-called directed polymer in random environment, being defined as follows: Let p(x, y) = p(y-x), x, y is an element of Z(d) be a shift-invarient, irreducible transition kernel, (S-n)(nis an element ofNo) the corresponding random walk. Let xi(x, n), x is an element of Z(d), n is an element of N be i.i.d. random variables satisfying E[exp(betaxi(x,n))] < ∞ for all β ∈ R, (1) We denote their cumulant generating function by λ(β) := log E[exp(βξ(x, n))]. (2) We think of the graph of S-n as the (directed) polymer, which is influenced by the random environment generated by the ξ(x, n) through a reweighting of paths with e(n) := e(n) (ξ, S) :=exp (Σ(n)(j=1) {betaxi(S-j,j) - gimel(beta)}), that is, we are interested in the random probability measures on path space given by mu(n) (d(s)) = 1/Z(n)E[e(n)1(S is an element of ds) \ xi(.,.)], where the normalising constant (or partition function) is given by [GRAPHICS] Note that (Z(n)) is a martingale, and hence converges almost surely. This model has been studied by many authors, see e.g. [2] and the references given there. It is known that the behaviour of mu(n) as n --> infinity depends on whether lim(n) Z(n) > 0 or limn Z(n) = 0. One speaks of weak disorder in the first, and of strong disorder in the second case. Our aim here is to give a condition for the weak disorder regime. Let (S-n) and S'(n)) be two independent p-random walks starting from S-0 = S'(0) = 0, and let V := Sigma(n=1)(infinity) 1(sn = S'n) be the number of times the two paths meet. Define alpha(*) := sup {alpha greater than or equal to 1 : E[alpha(V)\S'] < ∞ almost surely}.