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We study a random walk pinning model, where conditioned on a simple randomwalk Y on Z^d acting as a random medium, the path measure of a secondindependent simple random walk X up to time t is Gibbs transformed withHamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time between X andY up to time t. This model arises naturally in various contexts, including thestudy of the parabolic Anderson model with moving catalysts, the parabolicAnderson model with Brownian noise, and the directed polymer model. It falls inthe same framework as the pinning and copolymer models, and exhibits alocalization-delocalization transition as the inverse temperature \beta varies.We show that in dimensions d=1,2, the annealed and quenched critical values of\beta are both 0, while in dimensions d\geq 4, the quenched critical value of\beta is strictly larger than the annealed critical value (which is positive).This implies the existence of certain intermediate regimes for the parabolicAnderson model with Brownian noise and the directed polymer model. For d\geq 5,the same result has recently been established by Birkner, Greven and denHollander via a quenched large deviation principle. Our proof is based on afractional moment method used recently by Derrida, Giacomin, Lacoin andToninelli to establish the non-coincidence of annealed and quenched criticalpoints for the pinning model in the disorder-relevant regime. The critical cased=3 remains open.

Annealed vs quenched critical points for a random walk pinning model