Percolation thresholds and Fisher exponents in hypercubic lattices

Interplay of the complete-graph and Gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension

For statistical mechanical systems with continuous phase transitions, there are two closely related but subtly different mean-field treatments, the Gaussian fixed point (GFP) in the renormalization group framework and the Landau mean-field theory or the complete-graph (CG) asymptotics. By large-scale Monte Carlo simulations, we systematically study the interplay of the GFP and CG effects to the finite-size scaling of percolation above the upper critical dimension d_{c}=6 with periodic and cylindrical boundary conditions. Our results suggest that, with periodic boundaries, the unwrapped correlation length scales as L^{d/6} at the critical point, diverging faster than L above d_{c}. As a consequence, the scaling behaviors of macroscopic quantities with respect to the linear system size L follow the CG asymptotics. The distance-dependent properties, such as the short-distance behavior of the two-point correlation function and the Fourier transformed quantities with nonzero modes, are still controlled by the GFP. With cylindrical boundaries, due to the interplay of the GFP and CG effects, the correlation length along the axial direction of the cylinder scales as ξ_{L}∼L^{(d-1)/5} within the critical window of size O(L^{-2(d-1)/5}), distinct from periodic boundary. A field-theoretical calculation for deriving the scaling of ξ_{L} is also presented. Moreover, the one-point surface correlation function along the axial direction of the cylinder is observed to scale as τ^{(1-d)/2} when the distance τ is short, but then enter a plateau of order L^{-3(d-1)/5} before it decays significantly fast.

What are the limits of universality?

It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents on Z 4 and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent τ . This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A 382 , 12–19 ( doi:10.1016/j.physleta.2017.10.035 )).

High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas

The random Lorentz gas (RLG) is a minimal model for transport in disordered media. Despite the broad relevance of the model, theoretical grasp over its properties remains weak. For instance, the scaling with dimension d of its localization transition at the void percolation threshold is not well controlled analytically nor computationally. A recent study [Biroli et al., Phys. Rev. E 103, L030104 (2021)2470-004510.1103/PhysRevE.103.L030104] of the caging behavior of the RLG motivated by the mean-field theory of glasses has uncovered physical inconsistencies in that scaling that heighten the need for guidance. Here we first extend analytical expectations for asymptotic high-d bounds on the void percolation threshold and then computationally evaluate both the threshold and its criticality in various d. In high-d systems, we observe that the standard percolation physics is complemented by a dynamical slowdown of the tracer dynamics reminiscent of mean-field caging. A simple modification of the RLG is found to bring the interplay between percolation and mean-field-like caging down to d=3.

Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of D_{n} root lattices in n dimensions as well as E_{8}-related lattices. Here, we consider the percolation problem on D_{n} for n=3 to 13 and on E_{8} relatives for n=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_{n} lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as n increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.

Bond percolation on simple cubic lattices with extended neighborhoods

We study bond percolation on the simple cubic lattice with various combinations of first, second, third, and fourth nearest neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number z quite accurately according to a power-law p_{c}∼z^{-a} with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result p_{c}=1/(z-1). Fitting our data and data for additional nearest neighbors, we find p_{c}(z-1)=1+1.224z^{-1/2}.

Bond percolation between k separated points on a square lattice

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞, p[over ¯]_{c1}=0.51755(5), p[over ¯]_{c2}=0.53219(5), p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L^{-1/ν_{k}} where ν_{k}=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν_{1}=48/41, ν_{2}=24/23, ν_{3}=16/17, ν_{4}=6/7, etc. We also study three-point correlations in the system and show how for p>p_{c}, the correlation ratio goes to 1 (no net correlation) as L→∞, while at p_{c} it reaches the known value of 1.022.

Multirange Ising model on the square lattice

We study the Ising model on the square lattice (Z^{2}) and show, via numerical simulation, that allowing interactions between spins separated by distances 1 and m (two ranges), the critical temperature, T_{c}(m), converges monotonically to the critical temperature of the Ising model on Z^{4} as m→∞. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of 1, m, and u (three ranges), with u a multiple of m; in this case our results indicate that T_{c}(m,u) converges to the critical temperature of the model on Z^{6}. For percolation, analogous results were proven for the critical probability p_{c} [B. N. B. de Lima, R. P. Sanchis, and R. W. C. Silva, Stochast. Process. Appl. 121, 2043 (2011)STOPB70304-414910.1016/j.spa.2011.05.009].

Percolation of sites not removed by a random walker in d dimensions

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of N=uL^{d} steps on a d-dimensional hypercubic lattice of size L^{d} (with periodic boundaries). We systematically explore dependence of the probability Π_{d}(L,u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u. The concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated-their correlations decaying with the distance r as 1/r^{d-2} (in d>2). On increasing L, the percolation probability Π_{d}(L,u) approaches a step function, jumping from 1 to 0 when u crosses a percolation threshold u_{c} that is close to 3 for all 3≤d≤6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν=2/(d-2). There is no percolation threshold at the lower critical dimension of d=2, with the percolation probability approaching a smooth function Π_{2}(∞,u)>0.

Dynamics around the site percolation threshold on high-dimensional hypercubic lattices

Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, d_{u}=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥d_{u}. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below d_{u} as well as their logarithmic scaling above d_{u}. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.