We show that the limiting free energy in Sherrington-Kirkpatrick's Spin Glass Model does not depend on the environment.

Summary. We are interested in the randomly biased random walk on the supercritical Galton-Watson tree. Our attention is focused on a slow regime when the biased random walk (X n ) is null recurrent, making a maximal displacement of order of magnitude (log n) 3 in the first n steps. We study the localization problem of X n and prove that the quenched law of X n can be approximated by a certain invariant probability depending on n and the random environment. As a consequence, we establish that upon the survival of the system, |Xn| (log n) 2 converges in law to some nondegenerate limit on (0, ∞) whose law is explicitly computed.