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A scaling assumption for the numberg ns of different cluster configurations with perimeters and sizen leads to the desired cluster numbers near the percolation threshold. The perimeter distribution function has a mean square width proportional ton for largen. The relation between the average perimeter and the cluster sizen for percolation has three different forms atp c, belowp c, and abovep c and is closely related to the shape of the cluster size distribution.  相似文献   

3.
Renormalization group principles are used to argue that the Kunz-Souillard exponents are valid for all concentrations away from the percolation threshold, i.e. that the average numbers ns of clusters containing s sites each decay as log ns ∝ -sζ (s → ∞, p fixed), with ζ = 1 for all p below pc, and ζ = 1 - 1/d for all p above pc in d dimensions.  相似文献   

4.
We prove that the one-site distribution of Gibbs states (for any finite spin setS) on the Bethe lattice is given by the points satisfying the equation π=T 2π, whereT=h·A·?, with?(x)=x (q?1/q,h(x)=(xx q ) q ,A=(a(r, s)∶r, s∈S), and $$a(r,s) = \exp (K[r,s] + (1/q)[N,r + s])$$ We also show that forA a symmetric, irreducible operator the nonlinear evolution on probability vectorsx(n+1)=Ax(n) p Ax(n) p 1 withp>0 has limit pointsξ of period?2. We show thatA positive definite implies limit points are fixed points that satisfy the equation p=λξ. The main tool is the construction of a Liapunov functional by means of convex analysis techniques.  相似文献   

5.
Sang Bub Lee 《Physica A》2009,388(12):2271-2277
The mass distribution of invaded clusters in non-trapping invasion percolation between an injection site and an extraction site has been studied, in two, three, and four dimensions. This study is an extension of the recent study focused on two dimensions by Araújo et al. [A.D. Araújo, T.F. Vasconcelos, A.A. Moreira, L.S. Lucena, J.S. Andrade Jr., Phys. Rev. E 72 (2005) 041404] with respect to higher dimensions. The mass distribution exhibits a power-law behavior, P(m)∝mα. It has been found that the index α for pe<pc, pc being the percolation threshold of a regular percolation, appears to be independent of the value of pe and is also independent of the lattice dimensionality. When pe=pc, α appears to depend marginally on the lattice dimensionality, and the relation α=τ−1, τ being the exponent associated with cluster size distribution of a regular percolation via nssτ, appears to be valid.  相似文献   

6.
We show that the inverse correlation lengthm(β) (= mass of the fundamental particle of the associated lattice quantum field theory) of the spin-spin correlation function 〈s x s y 〉,x, y εZ d , of thed-dimensional Ising model admits the representation $$m(\beta ) = - ln\beta + r(\beta )$$ for small inverse temperaturesβ > 0.r(β) is ad-dependent function, analytic atβ = 0.c n , the nth β = 0 Taylor series coefficient of r(β) can be computed explicitly from the Zd limit of a finite number of finite lattice A spin-spin correlation functions 〈s0sx〉t>Afor a finite number ofx = (x 1,x2, ..., xd), ¦x¦ = ∑ i d 1¦xi¦< R(n), where R(n) increases withn. Furthermore, there exists aβ' > 0, such that for eachβ ε (0,β')m(β) is analytic. Similar results are also obtained for the dispersion curve ω(p), ω(p)=ω(0)=m, pε(-π, π]d?1, of the fundamental particle of the associated lattice quantum field theory.  相似文献   

7.
We study the iterations of the mapping $$\mathcal{N}[F(s)] = \frac{{(F(s))^2 - (F(0))^2 }}{s} + (F(0))^2 ,$$ with the constraintsF(1)=1,F(s)=∑a nsn,a n≧0, and find that, except ifF(s)≡s,N[F(s)] approaches either 0 or 1 for |s|<1 ask→∞.  相似文献   

8.
We consider bond percolation on $\mathbb{Z}^d$ at the critical occupation density p c for d>6 in two different models. The first is the nearest-neighbor model in dimension d?6. The second model is a “spread-out” model having long range parameterized by L in dimension d>6. In the spread-out case, we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as |x|→∞ when d>6 and L is sufficiently large. We also give a general criterion for this convergence to hold, which is satisfied in the case d?6 in the nearest-neighbor model by work of Hara.(12) We further give a second construction, by taking p<p c , defining a measure $\mathbb{Q}^p $ and taking its limit as pp ? c . The limiting object is the high-dimensional analogue of Kesten's incipient infinite cluster (IIC) in d=2. We also investigate properties of the IIC such as bounds on the growth rate of the cluster that show its four-dimensional nature. The proofs of both the existence and of the claimed properties of the IIC use the lace expansion. Finally, we give heuristics connecting the incipient infinite cluster to invasion percolation, and use this connection to support the well-known conjecture that for d>6 the probability for invasion percolation to reach a site x is asymptotic to c|x|?(d?4) as |x|→∞.  相似文献   

9.
We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, 2013, 87(5): 052107], it is observed that in comparison with dimensionless ratios based on cluster-size distribution, certain wrapping probabilities exhibit weaker finite-size corrections and are more sensitive to the deviation from percolation threshold Pc, and thus provide a powerful means for determining Pc. We analyze the numerical data of the wrapping probabilities simultaneously such that universal parameters are shared by the aforementioned models, and thus significantly improved estimates of Pc are obtained.  相似文献   

10.
The Coniglio-Stanley-Klein model is a random bond percolation process between the occupied sites of a lattice gas in thermal equilibrium. Our Monte Carlo simulation for 403 and 603 simple cubic lattices determines at which bond thresholdp Bc , as a function of temperatureT and concentrationx of occupied sites, an infinite network of active bonds connects occupied sites. The curvesp Bc (x, T) depend only slightly onT whereas they cross over if plotted as a function of the field conjugate tox. Except close toT c we find 1/p Bc to be approximated well by a linear function ofx, in the whole interval between the thresholdx c (T) of interacting site percolation atp Bc =1 and the random bond percolation limitx=1 atp Bc =0.248±0.001. Thisx c (T) varied between 0.22 forT=0.96 (coexistence curve) and 0.3117±0.0003 forT= (random site percolation). At the critical point (T=T andx=1/2) we confirmed quite accurately the predictionp Bc =1-exp(–2J/k B T c ) of Coniglio and Klein. As a byproduct we found 0.89±0.01 for the critical exponent of the correlation length in random percolation.  相似文献   

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