Universal Cluster Counting in the Tricritical Ising Model

Recent numerical studies of higher-order correlations in the critical Ising model have found discrepancies with field-theoretical predictions. One such study examined the number of clusters that intersect a given contour within a system. The count scales linearly with the contour length, with an additional universal logarithmic term originating from sharp corners in the contour, called the corner contribution. In the two-dimensional critical Ising model, analytic predictions for the corner contribution agree with numerical simulations for the Fortuin-Kasteleyn definition of clusters (FK clusters), but not for magnetic domains (spin clusters).

Through Monte Carlo methods, we investigate the corner contribution in the tricritical Ising model, or the Blume-Capel model. In contrast to the critical Ising model, we find that analytic predictions hold for the spin clusters but not for the FK clusters, supporting a conjectured analytic continuation between critical and tricritical Ising models. These observations highlight gaps in our theoretical understanding of the two-dimensional Ising model and motivate deeper investigation into higher-order correlations in critical systems.