Abstract
Birth-and-death processes or, equivalently, finite Markov chains with three-diagonal transition matrices proved to be adequate models for processes in physics [12], biology [4,5], sociology [13] and economics [1,3,10]. The analysis in this case quite often relies on the stationary distribution of the chain. Representing it as a Gibbs distribution, we study its limit behavior as the number of states increases. We show that the limit nests on the set of global minima of the limit Gibbs potential. If the set consists of a finite number k of singletons ai where the second derivatives ai of the potential are positive, the limit distribution assigns probability 1/√αi/∑j=1k 1/√αj to ai. When at some points the second derivative is zero, the limit distribution nests only on them, we describe it explicitly. If the set of minima consists of a finite number of singletons and intervals, the limit distribution concentrates only on intervals. We obtain a formula for it.