arXiv:cond-mat/9910071 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Large-qasymptotics of the random-bond Potts model

Jesper Lykke Jacobsen, Marco Picco

Published Jan 1, 2000• 215 citations

Abstract

We numerically examine the large-q asymptotics of the q-state random bond Potts model. Special attention is paid to the parametrization of the critical line, which is determined by combining the loop representation of the transfer matrix with Zamolodchikov's c-theorem. Asymptotically the central charge seems to behave like $c(q)=\frac{1}{2}{\mathrm{log}}_{2}(q)+O(1).$ Very accurate values of the bulk magnetic exponent ${x}_{1}$ are then extracted by performing Monte Carlo simulations directly at the critical point. As $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty},$ these seem to tend to a nontrivial limit, ${x}_{1}\ensuremath{\rightarrow}0.192\ifmmode\pm\else\textpm\fi{}0.002.$

DOI: 10.1103/physreve.61.r13

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arXiv:cond-mat/9910071 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Large-qasymptotics of the random-bond Potts model

Jesper Lykke Jacobsen, Marco Picco

Published Jan 1, 2000• 215 citations

arXiv HTML PDF

Abstract

c(q)=\frac{1}{2}{\mathrm{log}}_{2}(q)+O(1).

Very accurate values of the bulk magnetic exponent

{x}_{1}

are then extracted by performing Monte Carlo simulations directly at the critical point. As

\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty},

these seem to tend to a nontrivial limit,

{x}_{1}\ensuremath{\rightarrow}0.192\ifmmode\pm\else\textpm\fi{}0.002.

DOI: 10.1103/physreve.61.r13

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We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional $Q$ -state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents ${d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)$ and 1.794(2) for $Q=2,3$ , respectively. However, for large $Q$ , the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the $\mathrm{O}(n)$ loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with $Q={n}^{2}$ . With a highly efficient cluster algorithm, we determine from domains enclosed by the loops ${d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)$ for $Q=1,2,3,2+\sqrt{3},4$ , respectively, and ${d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)$ for $Q=2,3,2+\sqrt{3},4$ , respectively. Our estimates significantly improve over the existing results for both ${d}_{B}$ and ${d}_{\mathrm{min}}$ . Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of $n$ for the leading correction exponent.We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional $Q$-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents ${d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)$ and 1.794(2) for $Q=2,3$, respectively. However, for large $Q$, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the $\mathrm{O}(n)$ loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with $Q={n}^{2}$. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops ${d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)$ for $Q=1,2,3,2+\sqrt{3},4$, respectively, and ${d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)$ for $Q=2,3,2+\sqrt{3},4$, respectively. Our estimates significantly improve over the existing results for both ${d}_{B}$ and ${d}_{\mathrm{min}}$. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of $n$ for the leading correction exponent.

arXiv:2112.10162Physical review. EApr 15, 20229 citations

c(q)=\frac{1}{2}{\mathrm{log}}_{2}(q)+O(1).

Very accurate values of the bulk magnetic exponent

{x}_{1}

are then extracted by performing Monte Carlo simulations directly at the critical point. As

\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty},

these seem to tend to a nontrivial limit,

{x}_{1}\ensuremath{\rightarrow}0.192\ifmmode\pm\else\textpm\fi{}0.002.

(n=1)

, and others. In addition, for

q=8

, we find that there is only a single correlation length exponent $\ensuremath{\nu}$ describing the correlation length away from criticality. This is numerically very close to the pure Ising value $\ensuremath{\nu}=1$.

We present results of a numerical simulation of the

q

-state random bond Potts model in two dimensions and for large

q

. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.

We present a Monte Carlo study of the backbone and the shortest-path exponents of the two-dimensional

Q

-state Potts model in the Fortuin-Kasteleyn bond representation. We first use cluster algorithms to simulate the critical Potts model on the square lattice and obtain the backbone exponents

{d}_{B}=1.732\phantom{\rule{0.16em}{0ex}}0(3)

and 1.794(2) for

Q=2,3

, respectively. However, for large

Q

, the study suffers from serious critical slowing down and slowly converging finite-size corrections. To overcome these difficulties, we consider the

\mathrm{O}(n)

loop model on the honeycomb lattice in the densely packed phase, which is regarded to correspond to the critical Potts model with

Q={n}^{2}

. With a highly efficient cluster algorithm, we determine from domains enclosed by the loops

{d}_{B}=1.643\phantom{\rule{0.16em}{0ex}}39(5),1.732\phantom{\rule{0.16em}{0ex}}27(8),1.793\phantom{\rule{0.16em}{0ex}}8(3),1.838\phantom{\rule{0.16em}{0ex}}4(5),1.875\phantom{\rule{0.16em}{0ex}}3(6)

for

Q=1,2,3,2+\sqrt{3},4

, respectively, and

{d}_{\mathrm{min}}=1.094\phantom{\rule{0.16em}{0ex}}5(2),1.067\phantom{\rule{0.16em}{0ex}}5(3),1.047\phantom{\rule{0.16em}{0ex}}5(3),1.032\phantom{\rule{0.16em}{0ex}}2(4)

for

Q=2,3,2+\sqrt{3},4

, respectively. Our estimates significantly improve over the existing results for both

{d}_{B}

and

{d}_{\mathrm{min}}

. Finally, by studying finite-size corrections in backbone-related quantities, we conjecture an exact formula as a function of

n

for the leading correction exponent.