Singularity of full scaling limits of planar nearcritical percolation

We consider full scaling limits of planar nearcritical percolation in the Quad-Crossing-Topology introduced by Schramm and Smirnov. We show that two nearcritical scaling limits with different parameters are singular with respect to each other. The results hold for percolation models on rather general lattices, including bond percolation on the square lattice and site percolation on the triangular lattice.     

Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.     

Universality in two‐dimensional enhancement percolation

We consider a type of dependent percolation introduced in 2 , where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this study we first prove that, for two‐dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two‐dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit 12 , 13 is not affected by any monotonic enhancement that does not shift the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system

For dispersions containing a single type of particle, it has been observed that the onset of percolation coincides with a critical value of volume fraction. When the volume fraction is calculated based on excluded volume, this critical percolation threshold is nearly invariant to particle shape. The critical threshold has been calculated to high precision for simple geometries using Monte Carlo simulations, but this method is slow at best, and infeasible for complex geometries. This paper explores an analytical approach to the prediction of percolation threshold in polydisperse mixtures. Specifically, this paper suggests an extension of the concept of excluded volume, and applies that extension to the 2D binary disk system. The simple analytical expression obtained is compared to Monte Carlo results from the literature. The result may be computed extremely rapidly and matches key parameters closely enough to be useful for composite material design.     

Scaling Limit for the Ant in High-Dimensional Labyrinths

We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super-Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread-out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14. © 2019 Wiley Periodicals, Inc.     

On the connectivity properties of the complementary set in fractal percolation models

We study the connectivity properties of the complementary set in Poisson multiscale percolation model and in Mandelbrot's percolation model in arbitrary dimension. By using a result about majorizing dependent random fields by Bernoulli fields, we prove that if the selection parameter is less than certain critical value, then, by choosing the scaling parameter large enough, we can assure that there is no percolation in the complementary set.     

Critical percolation exploration path and SLE 6: a proof of convergence

It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE ₆. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE ₆ that we prove suffices for the Smirnov–Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops. Research of Charles M.Newman was partially supported by the US NSF under grants DMS-01-04278 and DMS-06-06696.     

Two-dimensional scaling limits via marked nonsimple loops

We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE₆ and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We showthat this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.     

Predicting dispersion in porous media

We present a fundamental theory of solute dispersion in porous using (i) critical path analysis and cluster statistics of percolation theory far from the percolation threshold and (ii) the tortuosity and structure of large clusters near the percolation threshold. We use the simplest possible model of porous media, with a single length scale of heterogeneity in which the statistics of local conductances are uncorrelated. This combination of percolation‐based techniques allows comprehensive investigation and predictions concerning the process of dispersion. Our predictions, which ignore molecular diffusion and make minimal use of unknown parameters, account for results obtained in a comprehensive set of nearly 1100 experiments performed on systems ranging in size from centimeters to 100 km. The success of our simple treatment overturns many existing notions about transport in porous media, such as (1) multiscale heterogeneity must be accounted for in predictions (single scale is sufficient), (2) geologic correlations are of great importance (the randomness of percolation theory is more appropriate for prediction than the most complicated models in other frameworks), (3) geologic complexity is more important than statistical physics (exactly the reverse), (4) knowledge of the subsurface is more important than knowledge of the initial conditions of the plume (the latter is critical, the former may be virtually irrelevant), (5) diffusion is dominant over advection (diffusion appears seldom to be relevant at all), (6) fracture networks are fundamentally different, and more complex, than porous media (the two are mostly equivalent), (7) the fractal structure of the medium is relevant to power‐law behavior of the dispersion (in fact, at short times it is the heterogeneity of the medium, while at long times it is the fractal structure of the critical paths), and (8) there is a relation between an increase in dispersion with scale and a similar increase in the hydraulic conductivity (in fact the present model is consistent with both a diminishing hydraulic conductivity and a diminishing solute velocity with increasing spatial scale). © 2009 Wiley Periodicals, Inc. Complexity, 16,43–55, 2010     

Critical behavior of percolation process influenced by a random velocity field: One-loop approximation

Using the perturbation renormalization group, we investigate the influence of a random velocity field on the critical behavior of the directed-bond percolation process near its second-order phase transition between the absorbing and active phases. We use the Antonov-Kraichnan model with a finite correlation time to describe the advecting velocity field. To obtain information about the large-scale asymptotic behavior of the model, we use the field theory renormalization group approach. We analyze the model near its critical dimension via a three-parameter expansion in ∈, δ, and η, where ∈ is the deviation from the Kolmogorov scaling, δ is the deviation from the critical space dimension, and η is the deviation from the parabolic dispersion law for the velocity correlator. We find the fixed points with the corresponding stability regions in the leading order in the perturbation scheme.     

低渗透介质含启动压力梯度一维瞬时压力分析

考虑启动压力梯度的影响,本文得到了在一维低渗透岩心和圆形油藏中一开一关渗流的数学方程,并进行有动边界问题的数值求解。以数值解为基础,分析了:1.渗流终态时井底(或岩心采液端)压力及压力影响半径(或距离)与启动压力梯度和关闭时间的关系,并给出相应的公式;2.压力峰值的传播规律。得到了有意义的结果,为低渗透油藏开发室内实验和现场应用提供理论基础。     

Site percolation and isoperimetric inequalities for plane graphs

We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.     

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.     

The water bag model and gyrokinetic applications

Predicting turbulent transport in nearly collisionless fusion plasmas requires to solve kinetic (or more precisely gyrokinetic) equations. In spite of considerable progress, several pending issues remain; although more accurate, the kinetic calculation of turbulent transport is much more demanding in computer resources than fluid simulations. An alternative approach is based on a water bag representation of the distribution function which is not an approximation but rather a special class of initial conditions allowing to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. This model has been applied to gyrokinetic modelling with very encouraging results. The instability threshold for ITG instability is found to be very close to the results obtained from continuous Maxwellian distribution, even for only 10 bags.     

An approximation for multi-server queues with deterministic reneging times

This work was motivated by the timeout mechanism used in managing application servers in transaction processing environments. In such systems, a customer who stays in the queue longer than the timeout period is lost. We modeled a server node with a timeout threshold as a multi-server queue with Poisson arrivals, general service time distribution and deterministic reneging times. We proposed a scaling approach, and a fast and accurate approximation for the expected waiting time in the queue.     

The coherent potential method in the diffusion problem on a random substitution lattice

Based on the balance equation, we consider the diffusion problem on a hyperlattice with randomly distributed inaccessible sites. Using diagram methods, we find a self-consistent expression for the configurationally averaged Green’s function in the coherent potential approximation. We show that this approach is applicable in a broad range of concentrations of accessible sites. Using this approximation, we find the exact asymptotic form of the static diffusion coefficient for a low concentration of blocked sites. This allows making good estimates of the percolation threshold in the random-site diffusion problem on an arbitrary hyperlattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 252–261, November, 2006.