Universality of Finite-Size Corrections to the Number of Critical Percolation Clusters

We present results for the high-temperature series expansions of the susceptibility and free energy of the q-state Potts model on a D-dimensional hypercubic lattice Z D for arbitrary values of q. The series are up to order 20 for dimension D ≤ 3, order 19 for D ≤ 5 and up to order 17 for arbitrary D. Using the q → 1 limit of these series, we estimate the percolation threshold pc and critical exponent γ for bond percolation in different dimensions. We also extend the 1/D expansion of the critical coupling for arbitrary values of q up to order D −9 .

Section: Mean Cluster Number and Moments Of The Cluster Numbersupporting

confidence: 69%

High-temperature series expansions for theq-state Potts model on a hypercubic lattice and critical properties of percolation

Hellmund

Janke

2006

“…Another interesting development for critical systems is finite-size corrections for lattice phase transition models [28,29,30,31,32,33]. Using exact partition functions of the Ising model on finite lattices [34] and exact finite-size corrections for the free energy, the internal energy, and the specific heat of the Ising model [32], Wu, Hu and Izmailian [35] obtained UFSSFs for the free energy, the internal energy, and the specific heat with analytic equations, which are free of simulation errors.…”

Section: Summary and Discussionmentioning

confidence: 99%

Mapping functions and critical behavior of percolation on rectangular domains

2008

The existence probability E p and the percolation probability P of the bond percolation on rectangular domains with different aspect ratios R are studied via the mapping functions between systems with different aspect ratios. The superscaling behavior of E p and P for such systems with exponents a and b, respectively, found by Watanabe, Yukawa, Ito, and Hu in [Phys. Rev. Lett. 93, 190601 (2004)] can be understood from the lower order approximation of the mapping functions f R and g R for E p and P , respectively; the exponents a and b can be obtained from numerically determined mapping functions f R and g R , respectively.

“…Using results from [5,6], Ref. [22] obtained explicit exact values of k c for bond percolation on three two-dimensional lattices:…”

Section: B Values Of K Cλmentioning

confidence: 99%

“…(Here we take into account that k is defined per site, while the quantity n B−HC c in [22] is defined per unit cell and there are two sites per unit cell on the honeycomb lattice.) Since p c,kag is not known exactly, neither is k c,kag .…”

Section: B Values Of K Cλmentioning

confidence: 99%

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Exact results for average cluster numbers in bond percolation on lattice strips

Chang¹,

Shrock²

2004

We present exact calculations of the average number of connected clusters per site, k , as a function of bond occupation probability p, for the bond percolation problem on infinite-length strips of finite width Ly, of the square, triangular, honeycomb, and kagomé lattices Λ with various boundary conditions. These are used to study the approach of k , for a given p and Λ, to its value on the two-dimensional lattice as the strip width increases. We investigate the singularities of k in the complex p plane and their influence on the radii of convergence of the Taylor series expansions of k about p = 0 and p = 1.