We consider a simple random walk of length , denoted by ( ), and we define ( ) a sequence of centered i.i.d. random variables. For ∈ℕ we define (( , …, )) an i.i.d sequence of random vectors. We set ∈ℝ, ≥0 and ≥0, and transform the measure on the set of random walk trajectories with the hamiltonian ∑ ( +)sign( )+∑ ∑ ...
We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed -dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically....
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