2026/07/15 by James Propp
#math.PR
This article associates to each positive real number x a dimer model on a snake graph with activity parameter q>0 whose structure is determined by the continued fraction expansion of x. When x is rational, the model is finite and gives rise to a probability measure μx,q on perfect matchings. For irrational x, the model is infinite, and μx,q is defined as a limit over rational approximations to x; the main technical result of the paper shows that this limit is well defined and independent of the choice of rational approximants. [[x]]q denotes the odds that a μx,q-random perfect matching includes a distinguished edge. When x is rational, [[x]]q = q [x]q, where [x]q is the algebraic q-deformation introduced by Morier-Genoud and Ovsienko. This agreement, together with evidence from the irrational case, suggests a close connection between the probabilistic and algebraic constructions.