Percolation in the canonical ensemble

Abstract

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y can = 2y t − d both in energy-like and magnetic quantities, where y t = 1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y can = −1/2.

Short-range correlations in percolation at criticality

“…As a result, the critical thermal fluctuations are suppressed in this model, so that the critical finite-size-scaling (FSS) amplitudes of many energy-like quantities vanish. For instance, the density of the occupied bonds is independent of the system size, and the density of clusters converges rapidly to its background value with zero amplitude for the leading finite-size term with y t = 1/ν in the exponent [8,9]. Though the partition function at q = 1 reduces to a trivial power of u, a number of nontrivial properties of the percolation model can be derived from the RC model via differentiation of the RC partition sum to q, and then taking the limit q → 1.…”

Universal critical wrapping probabilities in the canonical ensemble

“…The results summarized in the previous paragraph should apply but not restricted to these models, as long as the wrapping probability can be properly defined (wrapping is trivial on a complete graph). It should be mentioned that, for the percolation model, since the particle density has no finitesize dependence, the derivation presented need some modifications [7], which do not change the results for the wrapping probability. Exact values for the critical universal wrapping probability of the Potts model in the GCE were obtained through the analysis of the homology group of the torus based on a method introduced by di Francesco et al [38,2,8,9,10] (Ref.…”

“…We thank H. W. J. Blöte for the collaboration on Ref. [7] and a paper [39] which initiated this work, and for his critical reading of the manuscript. We also thank R. Ziff for his comments.…”

“…In a recent work [7], Hu et al studied the bond (site) percolation model with a fixed total number of occupied bonds (sites). The system is referred to as in the canonical ensemble (CE), while the model without the constraint is said to be in the grand-canonical ensemble (GCE).…”

Universal features of cluster numbers in percolation

“…A 0.0275981(3) [33] 0.8835(5) [33] 0.02759791(5) [19] 0.883576308... [37] 0.02759800(5) [39] 0.02759803 (2) [13] 0.017630(2) [40] 0.017626(1) [41] 0.0176255(5) [33] 0.878(1) [33] 0.017625277(4) m 0.87839(7) m 0.017625277368 (2) [13] 1.91392 (9) [13] 0.017788096(3) m 0.44183(1) m 0.017788106(1) s 0.441783154... [37] 0.01778810567665 . .…”