(the finite analog to the Bethe lattice) as where and the values of satisfy the recurrence relation In the case when the system is ferromagnetic, the above sequence converges, so we may take the limit to evaluate the magnetization on the Bethe lattice. We get where x is a solution to . There are either 1 or 3 solutions to this equation. In the case where there are 3, the sequence will converge to the smallest when and the largest when . Free energy The free energy f at each site of the lattice in the Ising Model is given by , where and is as before. In mathematics Return probability of a random walk The probability that a random walk on a Bethe lattice of degree starting at a given vertex eventually returns to that vertex is given by . To show this, let be the probability of returning to our starting point if we are a distance away. We have the recurrence relation for all , as at each location other than the starting vertex there are edges going away from the starting vertex and 1 edge going towards it. Summing this equation over all , we get . We have , as this indicates that we have just returned to the starting vertex, so , which is the value we want. Note that this in stark contrast to the case of random walks on the two-dimensional square lattice, which famously has a return probability of 1. Such a lattice is 4-regular, but the 4-regular Bethe lattice has a return probability of 1/3. Number of closed walks One can easily bound the number of closed walks of length starting at a given vertex of the Bethe Lattice with degree from below. By considering each step as either an outward step (away from the starting vertex) or an inward step (toward the starting vertex), we see that any closed walk of length must have exactly outward steps and inward steps. We also may not have taken more inward steps than outward steps at any point, so the number of sequences of step directions (either inward or outward) is given by the th Catalan number . There are at least choices for each outward step, and always exactly 1 choice for each inward step, so the number of closed walks is at least . This bound is not tight, as