Geometry, thermodynamics, and finite-size corrections in the critical Potts model

We establish an intriguing connection between geometry and thermodynamics in the critical q-state Potts model on two-dimensional lattices, using the q-state bond-correlated percolation model (QBCPM) representation. We find that the number of clusters $〈N_c〉$ of the QBCPM has an energylike singularity for $q\ne 1,$ which is reached and supported by exact results, numerical simulation, and scaling arguments. We also establish that the finite-size correction to the number of bonds, $〈N_b〉,$ has no constant term and explains the divergence of related quantities as $\overrightarrow{q}4,$ the multicritical point. Similar analyses are applicable to a variety of other systems.