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تسجيل مستخدم جديدThe Harmonic Measure for critical Potts clusters
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We present a technique, which we call etching, which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10^(-4600). We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q), including regions of small and negative q.
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We obtain the harmonic measure of the hulls of critical percolation clusters and Ising-model Fortuin-Kastelyn clusters using a biased random-walk sampling technique which allows us to measure probabilities as small as 10^{-300}. We find the multifrac
tal D(q) spectrum including regions of small and negative q. Our results for external hulls agree with Duplantiers theoretical predictions for D(q) and his exponent -23/24 for the harmonic measure probability distribution. For the complete hull, we find the probability decays with an exponent of -1 for both systems.
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urve P_B(q,v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = exp(K)-1, defined on G. This curve predicts the phase diagram both in the ferromagnetic (v>0) and antiferromagnetic (v<0) regions. For larger bases B the approximations become increasingly accurate, and we conjecture that P_B(q,v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G, or for the Ising model (q=2) on any G, P_B(q,v) factorises for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P_B(q,v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker-Kadanoff phase. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F.Y. Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wus approach. The polynomial predictions are in excellent agreement with numerical computations. For v>0 the accuracy of the predicted critical coupling v_c is of the order 10^{-4} or 10^{-5} for the 6-edge basis, and improves to 10^{-6} or 10^{-7} for the largest basis studied (with 36 edges).
The $q$-state Potts model has stood at the frontier of research in statistical mechanics for many years. In the absence of a closed-form solution, much of the past efforts have focused on locating its critical manifold, trajectory in the parameter ${
q, e^J}$ space where $J$ is the reduced interaction, along which the free energy is singular. However, except in isolated cases, antiferromagnetic (AF) models with $J<0$ have been largely neglected. In this paper we consider the Potts model with AF interactions focusing on deducing its critical manifold in exact and/or closed-form expressions. We first re-examine the known critical frontiers in light of AF interactions. For the square lattice we confirm the Potts self-dual point to be the sole critical point for $J>0$. We also locate its critical frontier for $J<0$ and find it to coincide with a solvability condition observed by Baxter in 1982. For the honeycomb lattice we show that the known critical point holds for {all} $J$, and determine its critical $q_c = frac 1 2 (3+sqrt 5) = 2.61803$ beyond which there is no transition. For the triangular lattice we confirm the known critical point to hold only for $J>0$. More generally we consider the centered-triangle (CT) and Union-Jack (UJ) lattices consisting of mixed $J$ and $K$ interactions, and deduce critical manifolds under homogeneity hypotheses. For K=0 the CT lattice is the diced lattice, and we determine its critical manifold for all $J$ and find $q_c = 3.32472$. For K=0 the UJ lattice is the square lattice and from this we deduce both the $J>0$ and $J<0$ critical manifolds and find $q_c=3$ for the square lattice. Our theoretical predictions are compared with recent tensor-based numerical results and Monte Carlo simulations.
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