The sequence of random probability measures
that gives a path of length , $\frac{1}{n}$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment.
Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.

We consider a catalytic branching random walk on $\mathbb{Z}$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position ${M}_{n}$: For some constant $\alpha $, $\frac{{M}_{n}}{n}\to \alpha $ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for ${M}_{n}-\alpha n$ as $n$ goes to infinity.

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed $(1+d)$-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the $d$ 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....

In this paper, we study particular examples of the intertwining relation
Q_{t}Λ = ΛP_{t}
between two Markov semi-groups (P_{t}, t ≥ 0) defined respectively on (E,ε) and (F,F), via the Markov kernel
Λ: (E,ε) → (F,F).

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