The q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.The q-state Potts model can be formulated in geometric terms, with Fortuin-Kasteleyn (FK) clusters as fundamental objects. For vanishing external field, the phase transition of the model can be equivalently described as a percolation transition of FK clusters. In this work, we investigate numerically the percolation behaviour along the line of first-order phase transitions of the 3d 3-state Potts model in a non-vanishing external field and find that the percolation strength exhibits a discontinuity along the entire line. The endpoint is also a percolation point for the FK clusters, but the corresponding critical exponents are neither in the Ising nor in the random percolation universality class. (C) 2002 Published by Elsevier Science B.V.

Aug 6, 20014 citations

Percolation renormalisation and simulation studies of the potts model

Chin‐Kun Hu, Kit-Sing Mak, Chi-Ning Chen

Nuclear Physics B - Proceedings SupplementsSep 1, 19880 citations

Monte Carlo study of the square-lattice annealed Ising model on percolating clusters

P. D. Scholten, Miron Kaufman

Simulations of an Ising $q$ -state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for $q=2,$ there are two critical Ising lines meeting at $T=0$ at the four-state Potts critical point; (ii) for $2<q<~4,$ the Ising critical line meets the $q$ -state critical line and a line of first-order transitions at a bicritical point; (iii) for $q>4,$ the Ising critical line intersects a line of first-order transitions at a critical end point.Simulations of an Ising $q$-state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for $q=2,$ there are two critical Ising lines meeting at $T=0$ at the four-state Potts critical point; (ii) for $2<q<~4,$ the Ising critical line meets the $q$-state critical line and a line of first-order transitions at a bicritical point; (iii) for $q>4,$ the Ising critical line intersects a line of first-order transitions at a critical end point.

Physical review. B, Condensed matterJul 1, 19979 citations

Dynamics of clusters in the two-dimensional Potts model

Y. Gündüç, M. Aydin

arXiv:hep-lat/9605015Physica A Statistical Mechanics and its ApplicationsOct 1, 19963 citations

Percolation approach to phase transitions

Antonio Coniglio A

Nuclear Physics AJan 1, 200114 citations

Active lattice percolation: an apparently new universality class of percolation transitions

Vernon M. Hughes, David A. Huse

We investigate a class of dynamic percolation models on a lattice, in which connections are added and removed by a set of local stochastic rules. In particular, the removal of any connection that can result in one connected cluster becoming two (or more) separate clusters is strictly forbidden. These models include and are inspired by a model for local quantum error correction for a toric code studied by Chirame, et al., PRX Quantum 6, 030363 (2025), who identified a phase transition with first-order characteristics, such as bistability and a discontinuous order parameter. We show numerically that this transition is a hybrid percolation transition (HPT), having characteristics of a continuous percolation transition such as diverging critical clusters while at the same transition having discontinuities like a first-order transition. We then modify this model to be fully isotropic, showing that the HPT remains. In both of these models, the steady state on one side of the HPT is an absorbing state; by turning on additional isotropic local stochastic moves, we remove the absorbing state and strongly suppress or remove the first-order transition, leaving an apparently continuous percolation transition. Curiously, the numerically-observed percolation critical exponents do not change significantly between the models with hybrid transitions and the model with an apparently continuous transition. These exponents are significantly different from known percolation transitions, indicating that this may be a new universality class.We investigate a class of dynamic percolation models on a lattice, in which connections are added and removed by a set of local stochastic rules. In particular, the removal of any connection that can result in one connected cluster becoming two (or more) separate clusters is strictly forbidden. These models include and are inspired by a model for local quantum error correction for a toric code studied by Chirame, et al., PRX Quantum 6, 030363 (2025), who identified a phase transition with first-order characteristics, such as bistability and a discontinuous order parameter. We show numerically that this transition is a hybrid percolation transition (HPT), having characteristics of a continuous percolation transition such as diverging critical clusters while at the same transition having discontinuities like a first-order transition. We then modify this model to be fully isotropic, showing that the HPT remains. In both of these models, the steady state on one side of the HPT is an absorbing state; by turning on additional isotropic local stochastic moves, we remove the absorbing state and strongly suppress or remove the first-order transition, leaving an apparently continuous percolation transition. Curiously, the numerically-observed percolation critical exponents do not change significantly between the models with hybrid transitions and the model with an apparently continuous transition. These exponents are significantly different from known percolation transitions, indicating that this may be a new universality class.

cond-mat.stat-mechCross-listedarXiv:2610.04807Oct 6, 20260 citations

Simulations of an Ising

q

-state Potts model which is equivalent to the Ising model on annealed percolation clusters are used to determine the phase diagram of the model in two dimensions. Three topologically different phase diagrams are obtained: (i) for

q=2,

there are two critical Ising lines meeting at

T=0

at the four-state Potts critical point; (ii) for

2<q<~4,

the Ising critical line meets the

q

-state critical line and a line of first-order transitions at a bicritical point; (iii) for

q>4,

the Ising critical line intersects a line of first-order transitions at a critical end point.