A critical branching random walk in a d-dimensional spatial random medium (environment) is considered. It is said to be "persistent" if there is no loss of particle intensity in the large time limit. A critical dimension dc is shown to exist such that the system is persistent if d>dc and fails to be persistent if d<dc.

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Persistent URL dx.doi.org/10.1007/BF00534864
Journal Probability Theory and Related Fields
Dawson, D.A, & Fleischmann, K. (Klaus). (1985). Critical dimension for a model of branching in a random medium. Probability Theory and Related Fields, 70(3), 315–334. doi:10.1007/BF00534864