Analytic origin of the $1/\log(LM)$ finite-size correction to the four-state Potts spin-cluster crossing probability

At the marginal point $q=4$ of the two-dimensional Potts model, the probability that a spin cluster crosses an $L\times M$ rectangle converges to the exact multiple-SLE$_4$ (conformal-field-theory) value only slowly, with a correction of order $1/\log(LM)$ rather than a power law, a signature of the marginally irrelevant operator of the $c=1$ fixed point. Fukushima and Sakai (2019), who observed this numerically, posed the still-open problem of deriving this logarithmic correction analytically from the correlation function of the boundary-condition-changing operators that encode the crossing probability. The problem is open.

Background

Some of the sharpest exact statements about two-dimensional critical phenomena concern non-local geometric observables rather than ordinary correlation functions. A canonical example is the crossing probability: the probability that, for a critical lattice model on a rectangle, a cluster connects two opposite sides. For critical bond percolation Cardy (1992) derived an exact formula, obtained as a conformal field theory (CFT) boundary four-point function; it depends only on the rectangle's aspect ratio $r=L/M$.

The $q$-state Potts model places such results in a one-parameter family. Spins $\sigma_j\in\{1,\dots,q\}$ sit on the sites of a square lattice with ferromagnetic nearest-neighbor coupling $J$; for $q\le 4$ the model has a second-order transition at $J_c=\log(1+\sqrt q)$, described by a CFT whose central charge (the number characterizing the universality class) is $c=0$ for $q=1$ (percolation), $c=1/2$ for $q=2$ (Ising), $c=4/5$ for $q=3$, and $c=1$ for $q=4$, the marginal case. One distinguishes Fortuin-Kasteleyn (FK) clusters, connected components in the random-cluster bond representation, from spin clusters, connected components of sites carrying the same spin value. Interfaces (cluster boundaries) are expected to converge, in the scaling limit, to a Schramm-Loewner evolution (SLE): a random conformally invariant curve driven by a one-dimensional Brownian motion, characterized by a single parameter $\kappa$ related to the central charge by $c=(3\kappa-8)(6-\kappa)/(2\kappa)$. For the four-state Potts model both the FK loops and - with the boundary prescription below - the spin-cluster interfaces correspond to $\kappa=4$. Since curves with $\kappa\le 4$ are simple and do not disconnect the domain, a crossing event involves several interfaces and cannot be extracted from a single curve.

For $\kappa\le 4$ the relevant object is a multiple SLE, a system of $N$ interacting curves started from boundary points, constructed by Bauer, Bernard and Kytölä (2005). The driving process contains a drift fixed by a partition function that is a correlation function of boundary-condition-changing (BCC) operators $\psi_{2,1}$ - operators inserted on the boundary where the boundary condition switches, degenerate at level two with conformal weight $h_{2,1}=(6-\kappa)/(2\kappa)$, which equals $1/4$ at $\kappa=4$. For $N=3$ curves started at $0<x<1$ on the real line, the two topologically distinct configurations have probabilities given by ratios of conformal blocks of this four-point function (the conformal blocks being the independent chiral solutions of the conformal Ward identities from which a CFT four-point function is assembled), and the Schwarz-Christoffel map - the explicit conformal map from the upper half-plane to a rectangle - to a rectangle of aspect ratio $r$ yields the scaling-limit crossing probability. At $\kappa=4$ it collapses to the closed form $P(r)=1-x$, where $x$ is fixed by $r=K(1-k^2)/(2K(k^2))$ with $k=(1-\sqrt x)/(1+\sqrt x)$ and $K$ the complete elliptic integral of the first kind.

On the lattice, matching spin-cluster interfaces to these curves requires care for $q\ge 3$, where several types of spin clusters coexist: with all boundary spins fixed, an interface starting at a corner splits into pieces of different types upon meeting clusters of other colors in the bulk. Gamsa and Cardy (2007) resolved this for $q=3$ using a fluctuating boundary condition, and Fukushima and Sakai (2019) did the same for $q=4$: the four spin values are grouped as $\{1,2\}$ and $\{3,4\}$, the top and bottom edges carry spins chosen randomly from $\{1,2\}$ while the left and right edges carry spins from $\{3,4\}$ (implemented as alternating $1212\dots$ and $3434\dots$ rows), and the interface is the boundary between the $\{1,2\}$ and $\{3,4\}$ clusters. Their Monte Carlo study confirmed numerically that this interface has the SLE$_4$ fractal dimension $d_f=1+\kappa/8=3/2$ (measured $1.504\pm0.020$) and that the probability of a $\{1,2\}$ cluster crossing from the top to the bottom edge is described by the $\kappa=4$ multiple-SLE formula in the scaling limit.

The subtlety lies in the approach to that limit. At $q=4$ the leading correction-to-scaling operator becomes marginally irrelevant: its conformal dimension vanishes, its coupling decays only logarithmically with system size, $g(L)\sim 1/\log L$, and observables acquire logarithmic corrections in place of power laws - quantitatively documented for bulk and strip observables of the two-dimensional four-state Potts model since the late 1980s, notably by Salas and Sokal (1997). Fukushima and Sakai found the same phenomenon in a boundary non-local observable: at fixed aspect ratio the finite-size data are well fitted by $P_{L\times M}(r)=c(r)+a(r)/\log(LM)$, where $LM$ is the number of lattice sites - e.g. $P=-0.2240/\log(LM)+0.9540$ at $r=0.55$ and $P=-0.2865/\log(LM)+0.8854$ at $r=0.65$ - and the deviation from the limiting value scales as $[1/\log(LM)]^{\,\approx 0.995}$, i.e. linearly in the inverse logarithm with a negative coefficient. Because the crossing probability is a ratio of correlation functions of the BCC operators, the natural analytic route is to perturb that four-point function by the marginally irrelevant operator of the $c=1$ fixed point. No such derivation exists; producing one is the problem stated here.

Problem Statement

Derive from first principles the finite-size scaling of the crossing probability of the critical four-state Potts model on the square lattice. Specifically: consider the model at $J=J_c=\log(1+\sqrt q)$, $q=4$, on an $L\times M$ rectangle with the fluctuating boundary condition in which the top and bottom edges carry spins chosen randomly from $\{1,2\}$ and the left and right edges spins from $\{3,4\}$, and let $P_{L\times M}(r)$, $r=L/M$, be the probability that a cluster of $\{1,2\}$ spins connects the top edge to the bottom edge. Its scaling limit is the multiple-SLE$_4$ result $P(r)=1-x$, where $x\in(0,1)$ is determined by $r=K(1-k^2)/(2K(k^2))$ with $k=(1-\sqrt x)/(1+\sqrt x)$ and $K$ the complete elliptic integral of the first kind. Monte Carlo data show $P_{L\times M}(r)=P(r)+A(r)/\log(LM)+o(1/\log(LM))$ with $A(r)<0$ at the measured aspect ratios, a logarithmic correction attributed to the marginally irrelevant operator of the $c=1$ fixed point.

The problem is to produce an analytic derivation of this asymptotic - starting from the $c=1$ boundary conformal field theory whose BCC operators $\psi_{2,1}$ of conformal weight $h=1/4$ enter the four-point function that encodes the crossing probability, perturbed by the marginally irrelevant operator - that (i) reproduces the scaling-limit formula $P(r)=1-x$; (ii) derives the leading correction of order $1/\log(LM)$, including its sign and its aspect-ratio dependence $A(r)$; and (iii) identifies which features of the correction are universal and which depend on the lattice realization. A derivation by another controlled route - for example an exact finite-lattice computation with the fluctuating boundary condition from which the same asymptotics follows - is equally acceptable, provided it yields the leading logarithmic term and the amplitude rather than only the scaling limit. As an explicitly secondary extension, determine whether the same mechanism produces analogous logarithmic corrections in other non-local geometric observables of the four-state Potts model (such as the FK-cluster boundary four-point connectivities - the probabilities that four given boundary points belong to the same Fortuin-Kasteleyn cluster - whose $Q=4$ finite-size behavior is likewise unresolved) and in neighboring $c=1$ conformal field theories; completion of the primary derivation constitutes the essential answer, and the extension probes the same mechanism in further observables. An answer that rigorously establishes a different asymptotic form - e.g. demonstrating that the observed slow convergence is not governed by $1/\log(LM)$ but by another mechanism - would equally resolve the question of its origin, provided it quantitatively accounts for the existing Monte Carlo data (the near-linear dependence of $P$ on $1/\log(LM)$ and the fitted exponents $\approx 0.99$ at $r=0.55$ and $r=0.65$).

References

1. Kimihiko Fukushima and Kazumitsu Sakai, 'Numerical study on a crossing probability for the four-state Potts model: Logarithmic correction to the finite-size scaling', Progress of Theoretical and Experimental Physics 2019, 091A01 (2019), DOI: 10.1093/ptep/ptz101, arXiv:1904.00406, https://arxiv.org/abs/1904.00406
2. Michel Bauer, Denis Bernard and Kalle Kytölä, 'Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales', Journal of Statistical Physics 120, 1125 (2005), DOI: 10.1007/s10955-005-7002-5, arXiv:math-ph/0503024, https://arxiv.org/abs/math-ph/0503024
3. J. Cardy, 'Critical percolation in finite geometries', Journal of Physics A: Mathematical and General 25, L201-L206 (1992), DOI: 10.1088/0305-4470/25/4/009, https://doi.org/10.1088/0305-4470/25/4/009
4. Adam Gamsa and John Cardy, 'Schramm-Loewner evolution in the three-state Potts model - a numerical study', Journal of Statistical Mechanics: Theory and Experiment 2007, P08020 (2007), DOI: 10.1088/1742-5468/2007/08/P08020, https://doi.org/10.1088/1742-5468/2007/08/P08020
5. J. Salas and A. D. Sokal, 'Logarithmic Corrections and Finite-Size Scaling in the Two-Dimensional 4-State Potts Model', Journal of Statistical Physics 88, 567 (1997), DOI: 10.1023/B:JOSS.0000015164.98296.85, https://doi.org/10.1023/B:JOSS.0000015164.98296.85
6. Giacomo Gori and Jacopo Viti, 'Exact Logarithmic Four-Point Functions in the Critical Two-Dimensional Ising Model', Physical Review Letters 119, 191601 (2017), DOI: 10.1103/PhysRevLett.119.191601, arXiv:1704.02893, https://arxiv.org/abs/1704.02893
7. Giacomo Gori and Jacopo Viti, 'Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance', Journal of High Energy Physics 2018, 131 (2018), DOI: 10.1007/JHEP12(2018)131, arXiv:1806.02330, https://arxiv.org/abs/1806.02330
8. Yoshiki Fukusumi, Marco Picco and Raoul Santachiara, 'Spin interfaces and crossing probabilities of spin clusters in parafermionic models' (2020), arXiv:2006.09850, https://arxiv.org/abs/2006.09850
9. Marco Picco, Sylvain Ribault and Raoul Santachiara, 'On four-point connectivities in the critical 2d Potts model', SciPost Physics 7, 044 (2019), DOI: 10.21468/SciPostPhys.7.4.044, arXiv:1906.02566, https://arxiv.org/abs/1906.02566
10. Victor Gorbenko and Bernardo Zan, 'Two-dimensional O(n) models and logarithmic CFTs', Journal of High Energy Physics 2020, 099 (2020), DOI: 10.1007/JHEP10(2020)099, arXiv:2005.07708, https://arxiv.org/abs/2005.07708
11. Federico Camia and Yu Feng, 'Logarithmic correlation functions in 2D critical percolation' (2024), arXiv:2403.18576, https://arxiv.org/abs/2403.18576
12. Gefei Cai, 'Boundary four-point connectivities of conformal loop ensembles' (2026), arXiv:2603.28161, https://arxiv.org/abs/2603.28161
13. Kimihiko Fukushima and Kazumitsu Sakai, 'Can a CNN trained on the Ising model detect the phase transition of the q-state Potts model?', Progress of Theoretical and Experimental Physics 2021, 061A01 (2021), DOI: 10.1093/ptep/ptab057, arXiv:2104.03632, https://arxiv.org/abs/2104.03632