Long-range Percolation Research Articles

How Fast do Rumours Spread?

We study a rumour propagation model along the lines of Lebensztayn and Rodriguez (Stat Probab Lett 78(14):2130–2136, 2008) as a long-range percolation model on Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}. We begin by showing a sharp phase transition-type behaviour in the sense of exponential decay of the survival time of the rumour cluster in the sub-critical phase. In the super-critical phase, under the assumption that radius of influence r.v. has 2+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2+\\epsilon $$\\end{document} moment finite (for some ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document}), we show that the rightmost vertex in the rumour cluster has a deterministic speed in the sense that after appropriate scaling, the location of the rightmost vertex converges a.s. to a deterministic positive constant. Under the assumption that radius of influence r.v. has 4+ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4+\\epsilon $$\\end{document} moment finite, we obtain a central limit theorem for appropriately scaled and centered rightmost vertex. Later, we introduce a rumour propagation model with reactivation. For this section, we work with a family of exponentially decaying i.i.d. radius of influence r.v.’s, and we obtain the speed result for the scaled rightmost position of the rumour cluster. Each of these results is novel, in the sense that such properties have never been established before in the context of the rumour propagation model on Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}, to the best of our knowledge.

Ultrahigh Responsivity and Robust Semiconducting Fiber Enabled by Molecular Soldering-Governed Defect Engineering for Smart Textile Optoelectronics.

Semiconducting fibers (SCFs) are of significant interest to design next-generation wearable and comfortable optoelectronics that seamlessly integrate with textiles. However, the practical applications of current SCFs are always limited by poor optoelectronic performance and low mechanical robustness caused by uncontrollable multiscale structural defects. Herein, a versatile in situ molecular soldering-governed defect engineering strategy is proposed to construct ultrahigh responsivity and robust wet-spun MoS2 SCFs, by using a π-conjugated dithiolated molecule to simultaneously patch microscale sulfur vacancies within MoS2 nanosheets, diminish mesoscale interlayer voids/wrinkles, promote macroscale orientation, build long-range photoelectron percolation bridges, and provide n-doping effect. The derived MoS2 SCFs exhibit over two orders of magnitude higher responsivity (144.3 A W-1) than previously reported fiber photodetectors, 37.3-fold faster photoresponse speed (52ms) than pristine counterpart, and remarkable bending robustness (retain 94.2% of the initial photocurrent after 50000 bending-flattening cycles). Such superior robustness and photodetection capacity of MoS2 SCFs further enable large-scale weaving of reliable smart textile optoelectronic systems, such as direction-identifiable wireless light alarming system, modularized mechano-optical communication system, and indoor light-controlled IoT system. This work offers a universal strategy for the scalable production of mechanically robust and high-performance SCFs, opening up exciting possibilities for large-scale integration of wearable optoelectronics.

Long-Range First-Passage Percolation on the Torus

We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {K}_n$$\\end{document} are embedded in the d-dimensional torus Tnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb T_n^d$$\\end{document}, and each edge e is assigned an independent transmission time Te=‖e‖TndαEe\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_e=\\Vert e\\Vert _{\\mathbb T_n^d}^\\alpha E_e$$\\end{document}, where Ee\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_e$$\\end{document} is a rate-one exponential random variable associated with the edge e, ‖·‖Tnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\cdot \\Vert _{\\mathbb T_n^d}$$\\end{document} denotes the torus-norm, and α≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\ge 0$$\\end{document} is a parameter. We are interested in the case α∈[0,d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in [0,d)$$\\end{document}, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document} studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the α=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =0$$\\end{document} case) studied by Janson [24]. We consider the typical distance, flooding time, and diameter of the model. Our results show a 1, 2, 3-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}^d$$\\end{document}.

Investigation of the fission gas release and grain boundary percolation in oxide fuels: A COMSOL multiphysics-based study

In this study, we explore fission gas release (FGR) behaviour in oxide fuels using the COMSOL Multiphysics framework. Building upon the conventional Booth model, our model incorporates the influence of grain boundary network percolation on FGR. The microstructure of the fuel pellet is represented by a 2D axi-symmetric geometry. We investigate various factors affecting FGR, mainly includes bubble contact angle, resolution rate, radial position. This study aims to capture the physical phenomena observed in FGR experiments and assess the sensitivity of FGR to these parameters at both microscopic and macroscopic scales. Moreover, our model accounts for long-range percolation on the networked grain boundaries, a factor overlooked by the Booth model. The gas resolution of grain boundaries is also considered. Finally, we compare and analyze the results obtained from our model with those from the Booth model and other relevant models, aiming at enhancing our understanding of FGR behavior in oxide fuels.

The longest edge in discrete and continuous long-range percolation

The longest edge in discrete and continuous long-range percolation

Arithmetic oscillations of the chemical distance in long-range percolation on Zd

Arithmetic oscillations of the chemical distance in long-range percolation on Zd

Inhomogeneous long-range percolation in the weak decay regime

We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the weak decay regime, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.

Isoperimetric lower bounds for critical exponents for long-range percolation

Isoperimetric lower bounds for critical exponents for long-range percolation

Double-exponential susceptibility growth in Dyson’s hierarchical model with |x − y|−2 interaction

We study long-range percolation on the d-dimensional hierarchical lattice, in which each possible edge {x, y} is included independently at random with inclusion probability 1 − exp(−β ‖x − y‖−d−α), where α > 0 is fixed and β ≥ 0 is a parameter. This model is known to have a phase transition at some βc < ∞ if and only if α < d. We study the model in the regime α ≥ d, in which βc = ∞, and prove that the susceptibility χ(β) (i.e., the expected volume of the cluster at the origin) satisfies χ(β)=βdα−d−o(1) as β↑∞ if α > d and χ(β)=eeΘ(β) as β↑∞ if α = d. This resolves a problem raised by Georgakopoulos and Haslegrave (2020), who showed that χ(β) grows between exponentially and double-exponentially when α = d. Our results imply that analogous results hold for a number of related models including Dyson’s hierarchical Ising model, for which the double-exponential susceptibility growth we establish appears to be a new phenomenon even at the heuristic level.

The critical two-point function for long-range percolation on the hierarchical lattice

The critical two-point function for long-range percolation on the hierarchical lattice

Cluster-size decay in supercritical long-range percolation

Cluster-size decay in supercritical long-range percolation

Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd

Consider long-range Bernoulli percolation on Zd in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(−β‖x − y‖−d−α), where α > 0 is fixed and β ⩾ 0 is a parameter. We prove that if 0 < α < d, then the critical two-point function satisfies 1|Λr|∑x∈ΛrPβc(0↔x)⪯r−d+α for every r ⩾ 1, where Λr=[−r,r]d∩Zd. In other words, the critical two-point function on Zd is always bounded above on average by the critical two-point function on the hierarchical lattice. This upper bound is believed to be sharp for values of α strictly below the crossover value αc(d), where the values of several critical exponents for long-range percolation on Zd and the hierarchical lattice are believed to be equal.

On the Derivation of Mean-Field Percolation Critical Exponents from the Triangle Condition

We give a new derivation of mean-field percolation critical behaviour from the triangle condition that is quantitatively much better than previous proofs when the triangle diagram \(\nabla _{p_c}\) is large. In contrast to earlier methods, our approach continues to yield bounds of reasonable order when the triangle diagram \(\nabla _p\) is unbounded but diverges slowly as \(p \uparrow p_c\), as is expected to occur in percolation on \({\mathbb {Z}}^d\) at the upper-critical dimension \(d=6\). Indeed, we show in particular that if the triangle diagram diverges polylogarithmically as \(p \uparrow p_c\) then mean-field critical behaviour holds to within a polylogarithmic factor. We apply the methods we develop to deduce that for long-range percolation on the hierarchical lattice, mean-field critical behaviour holds to within polylogarithmic factors at the upper-critical dimension. As part of the proof, we introduce a new method for comparing diagrammatic sums on general transitive graphs that may be of independent interest.

Percolation of words on the hypercubic lattice with one-dimensional long-range interactions

We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter 0 or 1 according to Bernoulli r.v.’s with parameter p. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on Zd−1×Z, d⩾3, where each edge parallel to Zd−1 has length one and is open with probability ϵ, while edges of length n parallel to Z are open with probability pn. We prove that if the sum of pn diverges, then for any ϵ and p, there is a K such that all words are seen from the origin with probability close to 1, even if all connections with length larger than K are suppressed.

Long-range contact process and percolation on a random lattice

We study the phase transition phenomena for long-range oriented percolation and contact process. We study a contact process in which the range of each vertex are independent, updated dynamically and given by some distribution N. We also study an analogous oriented percolation model on the hyper-cubic lattice, here there is a special direction where long-range oriented bonds are allowed; the range of all vertices are given by an i.i.d. sequence of random variables with common distribution N. For both models, we prove some results about the existence of a phase transition in terms of the distribution N.

Random walk on the random connection model

We study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices x and y are connected with probability that asymptotically behaves like |x−y|−α with α>d, where d denotes the dimension of the underlying Euclidean space. More precisely, focus is on the random connection model in which the vertex set is given by the realization of a homogeneous Poisson point process. We show that this random graph exhibits similar properties as classical discrete long-range percolation models studied by Berger (2002) with regard to recurrence and transience of the random walk. Moreover, we address a question which is related to a conjecture by Heydenreich, Hulshof and Jorritsma (2017) for this graph.

Cumulative merging percolation: A long-range percolation process in networks

Percolation on networks is a common framework to model a wide range of processes, from cascading failures to epidemic spreading. Standard percolation assumes short-range interactions, implying that nodes can merge into clusters only if they are nearest neighbors. Cumulative merging percolation (CMP) is a percolation process that assumes long-range interactions such that nodes can merge into clusters even if they are topologically distant. Hence, in CMP clusters do not coincide with the topologically connected components of the network. Previous work has shown that a specific formulation of CMP features peculiar mechanisms for the formation of the giant cluster and allows one to model different network dynamics such as recurrent epidemic processes. Here we develop a more general formulation of CMP in terms of the functional form of the cluster interaction range, showing an even richer phase transition scenario with competition of different mechanisms resulting in crossover phenomena. Our analytic predictions are confirmed by numerical simulations.

Large degrees in scale-free inhomogeneous random graphs

We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution.

Spectral dimension of simple random walk on a long-range percolation cluster

Consider the long-range percolation model on the integer lattice Zd in which all nearest-neighbour edges are present and otherwise x and y are connected with probability qx,y:=1−exp(−|x−y|−s), independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for s>d,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value d=1, s=2, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters s∈(d,2d). We further note that our approach is applicable to short-range models as well.

Truncation of long-range percolation models with square non-summable interactions

Truncation of long-range percolation models with square non-summable interactions