Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges c = -1 and c = -2. The first group includes generic p-regular maps with vertices of fixed valency p >= 3, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of 2q-regular maps (q >= 2) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent gamma , as well as a new universal critical exponent nu characterizing the number of longdistance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of nu, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for p-regular maps with p = 3, 4, ... , 7, and for maps with mixed valencies (2, 3), (2, 4) and (3, 4). (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3.