The Limit Distribution of a Number Theoretic Function Arising from a Problem in Statistical Mechanics

AbstractLetA≔1101,B≔1011, and for n∈N, let Φ(n) be the number of matrices C which are products of A's and B's where both A and B must occur, such that the trace tr(C)=n. It has been conjectured thatΦ(n)∼12π2nlogn,n→∞.This conjecture is disproved by showing thatφ(n)≔Φ(n)nlogn,n⩾2,has an absolutely continuous limit distribution with respect to the Lebesque measure