On a variational problem for an infinite particle system in a random medium Part II: The local growth rate - Greven, Andreas; den Hollander, Frank (Probability Theory and Related Fields, 1994)
This paper solves the second of two variational problems arising in the study of an infinite system of particles that branch and migrate in a random medium. This variational problem involves a non-linear functional on a subset of the stationary probability measures on [ℕ×ℝ], describing the interplay between particles and medium. It is shown that the variational problem can be solved in terms of the Lyapunov exponent of a product of random ℕ×ℕ matrices. This Lyapunov exponent is calculated via a random continued fraction. By analyzing the latter we are able to compute the maximum and the maximizer in the variational problem. It is found that these quantities exhibit interesting non-analyticities and changes of sign as a function of model parameters, which correspond to phase transitions in the infinite particle system. By combining with results from Part I we obtain a complete picture of the phase diagram. © 1994 Springer-Verlag.
Keywords: Mathematics Subject Classification 1991: 60F10; 60J15; 82B26; 82B44