Percolation threshold and critical point of a nuclear reactor

Abstract: Neutrons in a nuclear reactor move along trajectories corresponding to Cayley trees associated with branching random processes. The probability of percolation, the appearance of such a state of the Bethe lattice in which there is at least one continuous path through neighboring conducting nodes through the entire lattice, corresponds to the probability of the occurrence of a self-sustaining fission chain reaction. At a critical value of the last probability, a (conditionally) infinite cluster of neutrons appears. The probability of percolation, depending on the operating time of the reactor and its size, is associated with the criticality of the reactor. The temporal behavior of the neutron multiplication factor is estimated. Particular attention is paid to the early stages of the development of a self-sustaining chain reaction of nuclear fission. The possibilities of determining the boundaries of the critical region are indicated.