Two-dimensional Critical Percolation Research Articles

Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules

We study two-dimensional critical bootstrap percolation models. We establish that a class of these models including all isotropic threshold rules with a convex symmetric neighbourhood, undergoes a sharp metastability transition. This extends previous instances proved for several specific rules. The paper supersedes a draft by Alexander Holroyd and the first author from 2012. While it served a role in the subsequent development of bootstrap percolation universality, we have chosen to adopt a more contemporary viewpoint in its present form.

Logarithmic correlation functions in 2D critical percolation

It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities and to find an explanation and a physical interpretation, in terms of lattice observables, for their appearance. We show that certain percolation correlation functions receive independent contributions from a large number of similar connectivity events happening at different scales. Combined with scale invariance, this leads to logarithmic divergences. We study several logarithmic correlation functions for critical percolation in the bulk and in the presence of a boundary, including the four-point function of the density (spin) field. Our analysis confirms previous findings, provides new explicit calculations and explains, in terms of lattice observables, the physical mechanism that leads to the logarithmic singularities we discover. Although we adopt conformal field theory (CFT) terminology to present our results, the core of our analysis relies on probabilistic arguments and recent rigorous results on the scaling limit of critical percolation and does not assume a priori the existence of a percolation CFT. As a consequence, our results provide strong support for the validity of a CFT description of critical percolation and a step in the direction of a mathematically rigorous formulation of a logarithmic CFT of two-dimensional critical percolation.

An estimate for the radial chemical distance in [formula omitted] critical percolation clusters

We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length 2n, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path γ from the origin to distance n that consists of “three-arm” points, and whose volume can thus be estimated by O(n2π3(n)). Here, π3(n) is the “three-arm probability” that the origin is connected to distance n by three arms, two open and one dual-closed. We then develop estimates for the existence of shortcuts around an edge e in the box, conditional on {e∈γ}, to obtain a bound of the form O(n2−δπ3(n)) for some δ>0.

Chemical Subdiffusivity of Critical 2D Percolation

We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (Probab Theory Related Fields 73(3): 369–394 1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multiscale way of covering the graph so that “deep patches” have “thin backbones.” Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents \(\eta _2> \eta _1 > 0\): For d-dimensional models, the mean chemical displacement after T steps of random walk scales asymptotically slower than \(T^{1/\beta }\), whenever $$\begin{aligned} \beta < 2 + \frac{\eta _2-\eta _1}{d-\eta _1}\,. \end{aligned}$$Using the conjectured values of \(\eta _2 = \eta _1 + 1/4\) and \(\eta _1 = 5/48\) for 2D lattices, the latter quantity is \(2+12/91\).

Entanglement Negativity at Measurement-Induced Criticality

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their "mutual negativity" and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of "hybrid" circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension 3 across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

Two-point connectivity of two-dimensional critical Q-Potts random clusters on the torus

We consider the two dimensional Q-random-cluster Potts model on the torus and at the critical point. We study the probability for two points to be connected by a cluster for general values of . Using a conformal field theory (CFT) approach, we provide the leading topological corrections to the plane limit of this probability. These corrections have universal nature and include, as a special case, the universality class of two-dimensional critical percolation. We compare our predictions to Monte Carlo measurements. Finally, we take Monte Carlo measurements of the torus energy one-point function that we compare to CFT computations.

On the chemical distance in critical percolation

We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero in probability. This suggests that the chemical distance in critical percolation is given by an exponent strictly smaller than that of the lowest path.

Optimal transport exponent in spatially embedded networks

The imposition of a cost constraint for constructing the optimal navigation structure surely represents a crucial ingredient in the design and development of any realistic navigation network. Previous works have focused on optimal transport in small-world networks built from two-dimensional lattices by adding long-range connections with Manhattan length r(ij) taken from the distribution P(ij)~r(ij)(-α), where α is a variable exponent. It has been shown that, by introducing a cost constraint on the total length of the additional links, regardless of the strategy used by the traveler (independent of whether it is based on local or global knowledge of the network structure), the best transportation condition is obtained with an exponent α=d+1, where d is the dimension of the underlying lattice. Here we present further support, through a high-performance real-time algorithm, on the validity of this conjecture in three-dimensional regular as well as in two-dimensional critical percolation clusters. Our results clearly indicate that cost constraint in the navigation problem provides a proper theoretical framework to justify the evolving topologies of real complex network structures, as recently demonstrated for the networks of the US airports and the human brain activity.

Corner contribution to percolation cluster numbers

We study the number of clusters in two-dimensional ($2d$) critical percolation, ${N}_{\ensuremath{\Gamma}}$, which intersect a given subset of bonds, $\ensuremath{\Gamma}$. In the simplest case, when $\ensuremath{\Gamma}$ is a simple closed curve, ${N}_{\ensuremath{\Gamma}}$ is related to the entanglement entropy of the critical diluted quantum Ising model, in which $\ensuremath{\Gamma}$ represents the boundary between the subsystem and the environment. Due to corners in $\ensuremath{\Gamma}$ there are universal logarithmic corrections to ${N}_{\ensuremath{\Gamma}}$, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large-scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.

-extended fusion algebra of critical percolation

Two-dimensional critical percolation is the member of the infinite series of Yang–Baxter integrable logarithmic minimal models . We consider the continuum scaling limit of this lattice model as a ‘rational’ logarithmic conformal field theory with extended symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 -indecomposable representations with eight rank-1 representations, 14 rank-2 representations and four rank-3 representations. We identify these representations with suitable limits of Yang–Baxter integrable boundary conditions on the lattice and obtain their associated -extended characters. The latter decompose as finite non-negative sums of -irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.

Exact factorization of correlation functions in two-dimensional critical percolation

By use of conformal field theory, we discover several exact factorizations of higher-order density correlation functions in critical two-dimensional percolation. Our formulas are valid in the upper half-plane, or any conformally equivalent region. We find excellent agreement of our results with high-precision computer simulations. There are indications that our formulas hold more generally.

The density of critical percolation clusters touching the boundaries of strips andsquares

We consider the density of two-dimensional critical percolation clusters, constrained totouch one or both boundaries, in infinite strips, half-infinite strips and squares, aswell as several related quantities for the infinite strip. Our theoretical resultsfollow from conformal field theory and are compared with high-precision numericalsimulation. For example, we show that the density of clusters touching bothboundaries of an infinite strip of unit width (i.e. crossing clusters) is proportional to(sinπy)−5/48{[cos(πy/2)]1/3+[sin(πy/2)]1/3−1}. We also determine numerically contours for the density of clusters crossing squares andlong rectangles with open boundaries on the sides and compare with theory for the densityalong an edge.

Two-Dimensional Critical Percolation: The Full Scaling Limit

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.

Proposal for a conformal field theory interpretation of Watts’ differential equation forpercolation

G M T Watts established that in two-dimensional critical percolation the crossing probabilityΠhv satisfies a fifth-order differential equation which includes another one of thirdorder whose independent solutions describe the physically relevant quantities1,Πh,Πhv.We will show that this differential equation can be derived from a level three null vector condition for arational c = −24 conformal field theory and motivate how this solution may be fitted into known propertiesof percolation.

The lowest crossing in two-dimensional critical percolation

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.

Values of Brownian intersection exponents, II: Plane exponents

Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For instance, by making the assumption that critical planar percolation behaves in a conformally invariant way in the scaling limit, and using ideas involving conformal field theory, Cardy [7] produced an exact formula for the limit, as N → ∞, of the probability that, in two-dimensional critical percolation, there exists a cluster crossing the rectangle [0, aN] × [0, bN]. Also, Duplantier and Saleur [13] predicted the “fractal dimension” of the hull of a very large percolation cluster. These are just two examples among many such predictions.

Values of Brownian intersection exponents, I: Half-plane exponents

Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For instance, by making the assumption that critical planar percolation behaves in a conformally invariant way in the scaling limit, and using ideas involving conformal field theory, Cardy [7] produced an exact formula for the limit, as N → ∞, of the probability that, in two-dimensional critical percolation, there exists a cluster crossing the rectangle [0, aN] × [0, bN]. Also, Duplantier and Saleur [13] predicted the “fractal dimension” of the hull of a very large percolation cluster. These are just two examples among many such predictions.

Different self-avoiding walks on percolation clusters: A small-cell real-space renormalization-group study

We calculate the average number of stepsN for edge-to-edge, “normal,” and indefinitely growing self-avoiding walks (SAWs) on two-dimensional critical percolation clusters, using the real-space renormalization-group approach, with small “H” cells. Our results are of the formN=AL D SAW+B, whereL is the end-to-end distance. Similarly to several deterministic fractals, the fractal dimensionsD SAW for these three different kinds of SAWs are found to be equal, and the differences between them appear in the amplitudesA and in the correction termsB. This behavior is atributed to the hierarchical nature of the critical percolation cluster.

Violent fluctuations of the fracton density of states on the percolation cluster and its backbone.

We report the fracton density of states (DOS) and its fluctuation properties on the infinite two-dimensional critical percolation cluster and its backbone. The fracton DOS fluctuations, as expressed by the number variance 〈[\ensuremath{\delta}N(E)${]}^{2}$〉 in an energy width E, follow the quadratic law 〈[\ensuremath{\delta}N(E)${]}^{2}$〉\ensuremath{\propto}〈N(E)${\mathrm{〉}}^{2}$, instead of the usual linear Poissonic behavior normally expected for localized states. We also find that the average DOS for the percolation backbone obeys the one-parameter fracton scaling theory with a spectral dimension ${\mathit{d}}_{\mathit{s}}^{\mathrm{BB}}$=1.23\ifmmode\pm\else\textpm\fi{}0.02. This kind of violent DOS fluctuation cannot be understood in the context of random matrix theories and is discussed in connection to intermittency and multifractal localization.

Ideal chain on a two-dimensional critical percolation cluster

The authors study the statistics of the ideal chain (or equally weighted trajectories) for the first time on a two-dimensional critical percolation cluster. They discuss the asymptotic behaviour of the mean end-to-end distance and the number of chains for the long chain limit by exact enumeration. The results strongly suggest that this problem does not belong to the same universality class as the random walk (or kinetically weighted trajectories) on the same fractal cluster.