Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      2. A Note About Critical Percolation on Finite Graphs      Gady Kozma  Asaf Nachmias 《Journal of Theoretical Probability》2011,24(4):1087-1096 In this note we study the geometry of the largest component C1\mathcal {C}_{1} of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). There it is shown that this component is of size n 2/3, and here we show that its diameter is n 1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus \mathbbZnd\mathbb{Z}_{n}^{d} (with d large and n→∞) and the Hamming cube {0,1} n .      3. On Quantum Percolation in Finite Regular Graphs      Charles Bordenave 《Annales Henri Poincare》2015,16(11):2465-2497       4. Group-invariant solutions of the Fokker-Planck equation      G.A. Nariboli 《Stochastic Processes and their Applications》1977,5(2):157-171 Group-invariance under infinitesimal transformations is used to generate a wide class of solutions of some Fokker-Planck equations. The partial differential equation in two variables is reduced to an ordinary differential equation; reduction of the latter to standard forms is noted in most cases. Some of the known existing solutions are obtained as particular cases. Only self-similar types of solutions are discussed. The appearance of a free parameter that can be treated as an eigenvalue (or transform variable) offers flexibility in constructing new solutions. Some solutions of this parabolic equation have wave-like features. The general results can also be used to solve some types of moving-boundary problems.      5. Group-invariant solutions of a integrable coupled system      Suping Qian  Lixin Tian 《Nonlinear Analysis: Real World Applications》2008,9(4):1756-1767 In this paper, a new idea is put forward to modify the Clarkson–Kruskal's (CK's) direct method. By using the classical Lie group approach and the modified the CK's direct method, symmetry reductions and exact solutions are discussed for a integrable coupled KdV system. The group explanation for all the results obtained by the modified direct method is also given.      6. Group-invariant multiple solutions for quasilinear elliptic problems on strip-like domains      《Nonlinear Analysis: Theory, Methods & Applications》2013       7. Group-invariant analogues of Hadamard's inequality      Steen A. Anderson  Michael D. Perlman 《Linear algebra and its applications》1983 A wide class of inequalities for the determinant and other real-valued functions of an n × n complex Hermitian (or real symmetric) matrix H≡(hjk) may be obtained by generalizing Marshall and Olkin's proof of Hadamard's inequality $\text{det}\text{H?}\text{∏}\text{j=1}\text{n}\text{h}{}_{\mathrm{jj}}$ for positive definite (pd) H. We shall see that each subgroup G of the group Un of n x n unitary matrices not only determines an analogue of (1) for det H, but also provides inequalities for a large family of unitarily invariant functions of H (not necessarily pd).      8. Percolation diffusion      《Stochastic Processes and their Applications》2001,95(2):235-244       9. Sierpinski Gasket格点图上的渗流模型   总被引：2，自引：0，他引：2   吕建生 《数学进展》1999,28(6):519-526 本文研究了Ｓｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ上的边渗流模型。首先证明了该模型没有临界现象,进一步给出了其一个指数衰减律,进而证明了临界值的唯一性。      10. Percolation on random recursive trees          下载免费PDF全文 Erich Baur 《Random Structures and Algorithms》2016,48(4):655-680 We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree‐indexed process of cluster sizes to the genealogical tree of a continuous‐state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous‐time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 655–680, 2016      11. Energy of Flows on Percolation Clusters      Hoffman  Christopher  Mossel  Elchanan 《Potential Analysis》2001,14(4):375-385 It is well known for which gauge functions H there exists a flow in Z d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z d we let our (random) graph cal C cal (Z d,p) be the graph obtained from Z d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p c(Z d), simple random walk on cal C cal (Z d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x 2 energy e f(e)2 is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p c(Z d), then cal C cal (Z d,p) supports a nonzero flow f such that the x q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z d. In this paper we close the gap by showing that if d3 and Z d supports a flow of finite H energy then the infinite percolation cluster on Z d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.      12. 格点巢分形上的渗流模型      吕建生  Lü Jiansheng 《数学物理学报(A辑)》2000,20(4):521-527 该文研究了一类格点分形图 (格点巢分形 )上的渗流模型 ,证明了该模型没有临界现象 ,进一步给出一个指数衰减律 .同时 ,指出一般有限分岔图上的渗流模型没有临界现象      13. Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation          下载免费PDF全文 Jianping Shi  Mengmeng Zhou  Hui Fang 《Journal of Applied Analysis & Computation》2019,9(5):2023-2036 This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.      14. Group-invariant solutions for the Ricci curvature equation and the Einstein equation      Romildo Pina  João Paulo dos Santos 《Journal of Differential Equations》2019,266(4):2214-2231 We consider the pseudo-Euclidean space $({\mathbb{R}}^n,g)$, $n\ge 3$, with coordinates $x=(x_1,\dots ,x_n)$ and metric $g_{ij}={\delta }_{ij}{?}_i$, ${?}_i=\pm 1$, where at least one ${?}_i$ is positive, and also tensors of the form $A={\sum }_{i,j}{A}_{ij}d{x}_{i}d{x}_j$, such that $A_{ij}$ are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics $\stackrel{￣}{g}=\frac{1}{u^2}g$ that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics $\stackrel{￣}{g}$ defined globally in ${\mathbb{R}}^n$. As consequences, for certain functions $\stackrel{￣}{K}$, we show complete metrics $\stackrel{￣}{g}$, conformal to the pseudo-Euclidean metric g, whose scalar curvature is $\stackrel{￣}{K}$.      15. 关于紧图和超紧图的几个结果      周波  柳柏濂 《数学研究》1999,32(2):133-136 给出了一些 新的紧图,并对 不是超紧的紧图 作了一些讨论      16. Percolation on the three dot system      J.-F. Quint 《Probability Theory and Related Fields》2010,146(1-2):49-60 Let ${X\subset(\mathbb Z/2\mathbb Z)^{\mathbb Z^2}}$ be the three dot system. Given a ${\mathbb Z^2}$ -invariant ergodic probability measure on X, we study percolation properties on the set of 1’s in a typical orbit. This gives us a strong dichotomy for such measures.      17. Graphs on Stamps      Robin Wilson 《Mathematical Intelligencer》1990,12(1):80-80       18. Entanglement in Percolation      Grimmett  Geoffrey R.; Holroyd  Alexander E. 《Proceedings London Mathematical Society》2000,81(2):485-512 We study finite and infinite entangled graphs in the bond percolationprocess in three dimensions with density p. After a discussionof the relevant definitions, the entanglement critical probabilitiesare defined. The size of the maximal entangled graph at theorigin is studied for small p, and it is shown that this graphhas radius whose tail decays at least as fast as exp(–n/logn); indeed, the logarithm may be replaced by any iterate oflogarithm for an appropriate positive constant . We explorethe question of almost sure uniqueness of the infinite maximalopen entangled graph when p is large, and we establish two relevanttheorems. We make several conjectures concerning the propertiesof entangled graphs in percolation. http://www.statslab.cam.ac.uk/\simgrg/1991 Mathematics Subject Classification: primary 60K35; secondary05C10, 57M25, 82B41, 82B43, 82D60.      19. 一类奇异临界双调和椭圆方程的群不变解      邓志颖  黄毅生 《应用数学》2012,25(3):608-615 讨论一类含有Hardy-Sobolev临界指数项的奇异双调和椭圆方程,应用Lions集中紧性原理、Palais对称临界原理、Hardy-Rellich型不等式和变分方法,证明了方程在适当条件下群不变解的存在性和多重性.      20. 广义变系数Kdv方程的对称及其群不变解      蔡国梁  凌旭东  王庆超 《大学数学》2008,24(6) 利用经典李对称的方法对广义变系数Kdv方程进行研究,利用这种方法得到了该方程的一个新的精确解,这种方法的基本思路是通过对称约化将原来较难求解的偏微分方程转化为较易求解的常微分方程进行求解.实例证明这种方法具有一般性,适合于求一大类变系数的非线性演化方程.

A Note About Critical Percolation on Finite Graphs

In this note we study the geometry of the largest component C₁\mathcal {C}_{1} of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). There it is shown that this component is of size n ^2/3, and here we show that its diameter is n ^1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus \mathbbZ_n^d\mathbb{Z}_{n}^{d} (with d large and n→∞) and the Hamming cube {0,1} ⁿ .     

Group-invariant solutions of the Fokker-Planck equation

Group-invariance under infinitesimal transformations is used to generate a wide class of solutions of some Fokker-Planck equations. The partial differential equation in two variables is reduced to an ordinary differential equation; reduction of the latter to standard forms is noted in most cases. Some of the known existing solutions are obtained as particular cases. Only self-similar types of solutions are discussed. The appearance of a free parameter that can be treated as an eigenvalue (or transform variable) offers flexibility in constructing new solutions. Some solutions of this parabolic equation have wave-like features. The general results can also be used to solve some types of moving-boundary problems.     

Group-invariant solutions of a integrable coupled system

In this paper, a new idea is put forward to modify the Clarkson–Kruskal's (CK's) direct method. By using the classical Lie group approach and the modified the CK's direct method, symmetry reductions and exact solutions are discussed for a integrable coupled KdV system. The group explanation for all the results obtained by the modified direct method is also given.     

Group-invariant analogues of Hadamard's inequality

A wide class of inequalities for the determinant and other real-valued functions of an n × n complex Hermitian (or real symmetric) matrix H≡(h_jk) may be obtained by generalizing Marshall and Olkin's proof of Hadamard's inequality

$\text{det}\text{H?}\text{∏}\text{j=1}\text{n}\text{h}{}_{\mathrm{jj}}$

for positive definite (pd) H. We shall see that each subgroup G of the group U_n of n x n unitary matrices not only determines an analogue of (1) for det H, but also provides inequalities for a large family of unitarily invariant functions of H (not necessarily pd).     

Sierpinski Gasket格点图上的渗流模型

本文研究了Ｓｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ上的边渗流模型。首先证明了该模型没有临界现象,进一步给出了其一个指数衰减律,进而证明了临界值的唯一性。     

Percolation on random recursive trees

We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree‐indexed process of cluster sizes to the genealogical tree of a continuous‐state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous‐time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 655–680, 2016     

Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z ^d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z ^d we let our (random) graph cal C cal (Z ^d,p) be the graph obtained from Z ^d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p _c(Z ^d), simple random walk on cal C cal (Z ^d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x ² energy _e f(e)² is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p _c(Z ^d), then cal C cal (Z ^d,p) supports a nonzero flow f such that the x ^q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z ^d. In this paper we close the gap by showing that if d3 and Z ^d supports a flow of finite H energy then the infinite percolation cluster on Z ^d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.     

格点巢分形上的渗流模型

该文研究了一类格点分形图 (格点巢分形 )上的渗流模型 ,证明了该模型没有临界现象 ,进一步给出一个指数衰减律 .同时 ,指出一般有限分岔图上的渗流模型没有临界现象     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Group-invariant solutions for the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space $({\mathbb{R}}^n,g)$, $n\ge 3$, with coordinates $x=(x_1,\dots ,x_n)$ and metric $g_{ij}={\delta }_{ij}{?}_i$, ${?}_i=\pm 1$, where at least one ${?}_i$ is positive, and also tensors of the form $A={\sum }_{i,j}{A}_{ij}d{x}_{i}d{x}_j$, such that $A_{ij}$ are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics $\stackrel{￣}{g}=\frac{1}{u^2}g$ that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics $\stackrel{￣}{g}$ defined globally in ${\mathbb{R}}^n$. As consequences, for certain functions $\stackrel{￣}{K}$, we show complete metrics $\stackrel{￣}{g}$, conformal to the pseudo-Euclidean metric g, whose scalar curvature is $\stackrel{￣}{K}$.     

关于紧图和超紧图的几个结果

给出了一些 新的紧图,并对 不是超紧的紧图 作了一些讨论     

Percolation on the three dot system

Let ${X\subset(\mathbb Z/2\mathbb Z)^{\mathbb Z^2}}$ be the three dot system. Given a ${\mathbb Z^2}$ -invariant ergodic probability measure on X, we study percolation properties on the set of 1’s in a typical orbit. This gives us a strong dichotomy for such measures.     

Entanglement in Percolation

We study finite and infinite entangled graphs in the bond percolationprocess in three dimensions with density p. After a discussionof the relevant definitions, the entanglement critical probabilitiesare defined. The size of the maximal entangled graph at theorigin is studied for small p, and it is shown that this graphhas radius whose tail decays at least as fast as exp(–n/logn); indeed, the logarithm may be replaced by any iterate oflogarithm for an appropriate positive constant . We explorethe question of almost sure uniqueness of the infinite maximalopen entangled graph when p is large, and we establish two relevanttheorems. We make several conjectures concerning the propertiesof entangled graphs in percolation. http://www.statslab.cam.ac.uk/\simgrg/1991 Mathematics Subject Classification: primary 60K35; secondary05C10, 57M25, 82B41, 82B43, 82D60.     

一类奇异临界双调和椭圆方程的群不变解

讨论一类含有Hardy-Sobolev临界指数项的奇异双调和椭圆方程,应用Lions集中紧性原理、Palais对称临界原理、Hardy-Rellich型不等式和变分方法,证明了方程在适当条件下群不变解的存在性和多重性.     

广义变系数Kdv方程的对称及其群不变解

利用经典李对称的方法对广义变系数Kdv方程进行研究,利用这种方法得到了该方程的一个新的精确解,这种方法的基本思路是通过对称约化将原来较难求解的偏微分方程转化为较易求解的常微分方程进行求解.实例证明这种方法具有一般性,适合于求一大类变系数的非线性演化方程.