Richard P. Stanley Seminar in Combinatorics

Speaker: Jim Propp (UMass Lowell and Radcliffe Institute)

Title: Snake graphs and q-deformed real numbers

Abstract: I'll present a statistical-mechanical construction of q-deformed real numbers, inspired by the q-deformed rational numbers [x]q of Morier-Genoud and Ovsienko. The new construction admits two equivalent formulations: one in terms of random filters in snake posets and the other in terms of random dimer covers of snake graphs.

I'll begin with a combinatorial approach to Stern's diatomic series and the Calkin–Wilf bijection via finite snake graphs, and then construct q-deformed rational numbers by introducing an activity parameter q into the model. For irrational values of x, the construction requires passing from finite to infinite snake graphs; the main technical step is to show that the associated limiting probability measures are well defined.

The resulting q-deformation satisfies [[x]]1 = x for all x>0. There's evidence that, wherever both [x]q and [[x]]q are defined, [[x]]q = q [x]q. For more details, see arXiv:2607.14332.