Dimers, filters, and $q$-deformed real numbers

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 $\mu_{x,q}$ on perfect matchings. For irrational $x$, the model is infinite, and $\mu_{x,q}$ 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 $\mu_{x,q}$-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.