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In this paper we introduce and study renewal–reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments. [ABSTRACT FROM AUTHOR]

We consider a directed polymer of length N interacting with a linear interface. The monomers carry i.i.d. random charges (ω i) i = 1 N taking values in R with mean zero and variance one. Each monomer i contributes an energy (β ω i - h) φ (S i) to the interaction Hamiltonian, where S i ∈ Z is the height of monomer i with respect to the interface, φ : Z → [ 0 , ∞) is the interaction potential, β ∈ [ 0 , ∞) is the inverse temperature, and h ∈ R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on Z . We study both the quenched and the annealed free energy per monomer in the limit as N → ∞ . We show that each exhibits a phase transition along a critical curve in the (β , h) -plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as β , h ↓ 0 in three different regimes for the tail exponent of φ . [ABSTRACT FROM AUTHOR]

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established. [ABSTRACT FROM AUTHOR]