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Polymer pinning with sparse disorder

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The standard setup in disordered pinning models is that a polymer configuration is modeled by the trajectory of a Markov chain which is rewarded or penalized by an amount \omega_n when it returns to a special state 0 at time n. More precisely, for a polymer of length N the Boltzmann weight is e^{\beta H}, where for a trajectory \tau, H(\tau) is the sum of the values \omega_n at the times n \leq N of the returns to 0 of \tau. Typically the \omega_n are taken to be a quenched realization of a iid sequence, but here we consider the case of sparse disorder: \omega_n is 1 at the returns times of a quenched realization of a renewal sequence \{\sigma_j\}, and 0 otherwise; in the interesting cases the gaps between renewals have infinite mean, and we assume the gaps have a regularly varying power-law tail. For \beta above a critical point, the polymer is pinned in the sense that \tau asymptotically hits a positive fraction of the N renewals in \sigma. To see the effect of the disorder one can compare this critical point to the one in the corresponding annealed system. We establish equality or inequality of these critical points depending on the sum of the tail exponents of the two renewal sequences (that is, \sigma and the return times of \tau.) This is joint work with Quentin Berger.

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