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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 &lt;N(c)&gt; of the QBCPM has an energy-like singularity for q not equal to 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, &lt;N(b)&gt;, has no constant term and explains the divergence of related quantities as q--&gt;4, the multicritical point. Similar analyses are applicable to a variety of other systems.

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