A Harris‐Kesten theorem for confetti percolation

Percolation properties of the dead leaves model, also known as confetti percolation, are considered. More precisely, we prove that the critical probability for confetti percolation with square‐shaped leaves is 1/2. This result is related to a question of Benjamini and Schramm concerning disk‐shaped leaves and can be seen as a variant of the Harris‐Kesten theorem for bond percolation. The proof is based on techniques developed by Bollobás and Riordan to determine the critical probability for Voronoi and Johnson‐Mehl percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 361–385, 2015     

Is the critical percolation probability local?

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. We also prove a finite analogue of this statement, valid for expander graphs, without any girth assumption.     

A note on percolation

Summary An improvement of Harris' theorem on percolation is obtained; it implies relations between critical points of matching graphs of the type of the one stated by Essam and Sykes. As another consequence, it is proved that the percolation probability, as a function of the probability of occupation of a given site, is infinitely differentiable, except at most in the critical point.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

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     

Cut sets and normed cohomology with applications to percolation

We discuss an inequality for graphs, which relates the distances between components of any minimal cut set to the lengths of generators for the homology of the graph. Our motivation arises from percolation theory. In particular this result is applied to Cayley graphs of finite presentations of groups with one end, where it gives an exponential bound on the number of minimal cut sets, and thereby shows that the critical probability for percolation on these graphs is neither zero nor one. We further show for this same class of graphs that the critical probability for the coalescence of all infinite components into a single one is neither zero nor one.

     

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersley’s critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersley’s critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet.Most results of this paper are announced in Okamura (2016) [24] without proofs. This paper gives full details of them.     

二维双重定向渗流及其临界概率函数

本文研究二维双重定向渗流模型，我们给出模型临界概率函数的一些基本性质，包括严格单调性、对称性和连续性，另外，我们指出，该临界概率函数的严格凹性是Grimmett相关猜想的充分条件。     

Non-existence of phase transition of oriented percolation on Sierpinski carpet lattices

 A percolation problem on Sierpinski carpet lattices is considered. It is obtained that the critical probability of oriented percolation is equal to 1. In contrast it was already shown that the critical probability p _c of percolation is strictly less than 1 in Kumagai [9]. This result shows a difference between fractal-like lattice and ℤ ^d lattice. Received: 15 May 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): Primary: 60K35, 82B43; Secondary: 82B26     

Analysis,Simulation, and Optimization of Stochastic Vesicle Dynamics in Synaptic Transmission

Synaptic transmission is the mechanism of information transfer from one neuron to another (or from a neuron to a muscle or to an endocrine cell). An important step in this physiological process is the stochastic release of neurotransmitter from vesicles that fuse with the presynaptic membrane and spill their contents into the synaptic cleft. We are concerned here with the formulation, analysis, and simulation of a mathematical model that describes the stochastic docking, undocking, and release of synaptic vesicles and their effect on synaptic signal transmission. The focus of this paper is on the parameter p₀, the probability of release for each docked vesicle when an action potential arrives. We study the influence of this parameter on the statistics of the release process and on the theoretical capability of the model synapse in reconstructing various desired outputs based on the timing and amount of neurotransmitter release. This theoretical capability is assessed by formulating and solving an optimal filtering problem. Methods for parameter identification are proposed and applied to simulated data. © 2019 Wiley Periodicals, Inc.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Coexistence of the infinite (*) clusters: — A remark on the square lattice site percolation

Summary We show that the critical probability p _c is strictly greater than 1/2 for the square lattice site percolation.     

Functional motifs: a novel perspective on burst synchronization and regularization of neurons coupled via delayed inhibitory synapses

For one of the most common network motifs, an inhibitory neuron pair, we perform an extensive study of burst synchronization and the related phenomena applying the model of Rulkov maps coupled via delayed synapses. Instigated by the phase-plane analysis, that has the neuron switching between the noninteracting and the interacting map, it is demonstrated how the system evolution may be interpreted by means of the dynamical configurations of the motif, each represented by an extracted subgraph. Under the variation of the synaptic parameters, the probability of finding synchronized neurons in a given configuration is seen to reflect the way in which the anti-phase synchronization is eventually superseded by the synchronization in phase. Such an approach also provides a novel insight into regularization, characterizing the neuron bursting in either of these regimes. Looking into correlation of the two neurons’ bursting cycles we acquire a deeper understanding of the more sophisticated mechanisms by which the regularity in the time series is maintained. Further, it is examined whether introducing heterogeneity in the neuron or the synaptic parameters may prove advantageous over the homogeneous case with respect to burst synchronization.     

Sharp thresholds and percolation in the plane

Recently, it was shown by Bollobás and Riordan 4 that the critical probability for random Voronoi percolation in the plane is 1/2. As a by‐product of the method, a short proof of the Harris–Kesten Theorem was given by Bollobás and Riordan 5 . The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results and to prove new ones. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

A lower bound for the critical probability of the square lattice site percolation

Summary We prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0.503478.Work supported by the Central Research Found of the Hungarian Academy of Sciences (Grant No. 476/82)     

A new lower bound for the critical probability of site percolation on the square lattice

The critical probability for site percolation on the square lattice is not known exactly. Several authors have given rigorous upper and lower bounds. Some recent lower bounds are (each displayed here with the first three digits) 0.503 (Tóth [13]), 0.522 (Zuev [15]), and the best lower bound so far, 0.541 (Menshikov and Pelikh [12]). By a modification of the method of Menshikov and Pelikh we get a significant improvement, namely, 0.556. Apart from a few classical results on percolation and coupling, which are explicitly stated in the Introduction, this paper is self-contained. © 1996 John Wiley & Sons, Inc.     

The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm [5].     

Jamming in attractive granular media

The jamming transition is studied numerically in systems of particles with attraction. Unlike the purely repulsive case where a single transition separates the jammed from unjammed phase, the presence of even an infinitesimal amount of attraction yields two distinct transitions: connectivity and rigidity percolation. We measure critical exponents of these two percolation transitions and find that they are different than the corresponding lattice values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

时滞对磁通耦合及化学耦合神经元分岔及同步的影响

以化学突触耦合神经元模型为基础,讨论了抑制性及兴奋性条件下达到同步的区别及同步的类型。并根据磁通耦合对神经元放电的影响,讨论了具有时滞、磁通耦合和化学耦合Morris-Lecar (ML)神经元模型的放电状态、分岔类型及其同步情况。发现具有磁通耦合和化学耦合ML神经元系统在不同参数下会产生丰富的逆倍周期分岔或加周期分岔行为。而时滞的引入,虽然可以增加系统的周期性,但同时也会破环系统同步。相反,适当的耦合强度能够增加同步。     

Social percolation in relations between activists and supporters

This article is a study of the relations between activists with a strong belief and their passive supporters. In order to achieve a fuller understanding of these relations the authors make use of bond percolation theory. The theories hold that when the size of a group's communication in a given society exceeds a critical threshold; that is, when the size of the cluster of supporters' bonds becomes practically infinite within the society by virtue of the work of activists, subsequently supporters and activists come to hold key positions in society. A graph is generated indicating the probability of activists' bonds with respect to the probability of supporters' bonds within a lattice, when the biggest cluster of supporters' bonds becomes infinite. Simulations have been undertaken, and when a case is found in which infinite clusters emerge, this value of the probability of both supporters' and activists' bonds is isolated. A graph can then be generated a graph on which to plot these values as points. After that, these results can be discussed at length and some conclusions are derived from them. © 2006 Wiley Periodicals, Inc. Complexity 11: 51–56, 2006