Thermodynamics of a Brownian Bridge Polymer Model in a Random Environment 1

We consider a directed random walk making either 0 or +1 moves and a Brownian bridge, independent of the walk, conditionned to arrive at point b on time T. The Hamiltonian is deened as the sum of the square of increments of the bridge between the moments of jump of the random walk and interpreted as an energy function over the bridge connguration; the random walk acts as the random environment. The thermodynamic limit of the speciic free energy is shown to exist and to be self-averaging, i.e. it is equal to a trivial | explicitely computed | random variable. An estimate of the asymptotic behaviour of the ground state energy is also obtained. 1 Deenition of the model and main results Disordered systems are thoroughly studied nowadays as condensed matter models in the presence of impurities. Interesting questions, both mathematically and physically, concern the sample dependence or independence of intensive quantities like the speciic free energy in statistical mechanics of spin glasses or the integrated density of states in spectral theory