Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ?^d. That is, for each d ≥ 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias such that below this value the speed is positive, and above the value it is zero. We identify the value of the critical bias explicitly, and in the subballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one‐dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known. © 2013 Wiley Periodicals, Inc.     

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

We consider a continuum percolation model on \(\mathbb {R}^d\), \(d\ge 1\). For \(t,\lambda \in (0,\infty )\) and \(d\in \{1,2,3\}\), the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity \(\lambda >0\). When \(d\ge 4\), the Brownian paths are replaced by Wiener sausages with radius \(r>0\). We establish that, for \(d=1\) and all choices of t, no percolation occurs, whereas for \(d\ge 2\), there is a non-trivial percolation transition in t, provided \(\lambda \) and r are chosen properly. The last statement means that \(\lambda \) has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when \(d\in \{2,3\}\), but finite and dependent on r when \(d\ge 4\)). We further show that for all \(d\ge 2\), the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.     

Dynamical Phase Transition for a Quantum Particle Source

We analyze the time evolution describing a quantum source for non-interacting particles, either bosons or fermions. The growth behavior of the particle number (trace of the density matrix) is investigated, leading to spectral criteria for sublinear or linear growth in the fermionic case, but also establishing the possibility of exponential growth for bosons. We further study the local convergence of the density matrix in the long time limit and prove the semi-classical limit.     

Evidence for a Phase Transition in Three-Dimensional Lattice Models

It was recently discovered that an eigenvector structure of commutative families of layer-to-layer matrices in three-dimensional lattice models is described by a two-dimensional spin lattice generalizing the notion of one-dimensional spin chains. We conjecture the relations between the two-dimensional spin lattice in the thermodynamic limit and the phase structure of three-dimensional lattice models. We consider two simplest cases: the homogeneous spin lattice related to the Zamolodchikov–Bazhanov–Baxter model and a chess spin lattice related to the Stroganov–Mangazeev elliptic solution of the modified tetrahedron equation. Evidence for the phase transition is obtained in the second case.     

An Analogue of Slepian Vectors on Boolean Hypercubes

Journal of Fourier Analysis and Applications - Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the...     

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 ,}      9. Un Exemple de Transition de Phase      François Ledrappier 《Monatshefte für Mathematik》1977,83(2):147-153 We give a new proof of the non uniqueness of the equilibrium state inF. Hofbauer's example. We extend it to obtain examples with any finite number of ergodic equilibrium states.      10. Phase Transition Models in Atmospheric Dynamics      Arthur Bousquet  Michele Coti Zelati  Roger Temam 《Milan Journal of Mathematics》2014,82(1):99-128 Abstract From both the theoretical and numerical viewpoints, we study a system of differential inclusions describing the evolution of the temperature and the specific humidity distributions in a system of moist air. We allow the so-called saturation concentration parameter to depend on the temperature, and thus we consider more general and interesting phase-change effects than the ones addressed in [2].      11. A Phase Transition for Large Values of Bifurcating Autoregressive Models      Bansaye  Vincent  Bitseki Penda  S. Valère 《Journal of Theoretical Probability》2021,34(4):2081-2116 Journal of Theoretical Probability - We describe the asymptotic behavior of the number $$Z_n[a_n,infty )$$ of individuals with a large value in a stable bifurcating autoregressive process, where...      12. Group-invariant Percolation on Graphs      I. Benjamini  R. Lyons  Y. Peres  O. Schramm 《Geometric And Functional Analysis》1999,9(1):29-66 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.      13. Dependence of the Phase Transition Temperature on the Domain Size      V. G. Osmolovski 《Journal of Mathematical Sciences》2006,132(3):304-312 Dependence of the phase transition temperature on the domain size is investigated for a double-well quadratic potential. It is shown that for a domain whose boundary is subjected to a hydrostatical pressure, the temperature of phase transitions is independent of the domain and the surface tension coefficient and depends exclusively on the properties of the elastic media. If the displacement field vanishes on the boundary, then for sufficiently small domains, the temperature also does not depend on the surface tension and domain size and is determined by properties of the elastic media only. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 98–113.      14. Isotropic-Nematic Phase Transition and Liquid Crystal Droplets      Fanghua Lin  Changyou Wang 《纯数学与应用数学通讯》2023,76(9):1728-1792       15. Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices          下载免费PDF全文 Shirshendu Chatterjee  Partha S. Dey 《纯数学与应用数学通讯》2016,69(2):203-256 We consider a model of long‐range first‐passage percolation on the d‐dimensional square lattice ?d in which any two distinct vertices x,y ? ?d are connected by an edge having exponentially distributed passage time with mean ‖ x – y ‖α+o(1), where α > 0 is a fixed parameter and ‖·‖ is the l1–norm on ?d. We analyze the asymptotic growth rate of the set ßt, which consists of all x ? ?d such that the first‐passage time between the origin 0 and x is at most t as t → ∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α < d, (ii) stretched exponential growth for α ? d,2d), (iii) superlinear growth for α ? (2d,2d + 1), and finally (iv) linear growth for α > 2d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α=∞. © 2015 Wiley Periodicals, Inc.      16. Phase Transition in the Mechanics of Continuous Media for Big Loading      V. G. Osmolovskij 《Mathematische Nachrichten》1996,177(1):233-250 In this note a matrix partial differential operator is considered. It is shown that under certain conditions it defines a closed operator with nonempty resolvent set, and its essential spectrum is determined. In the symmetric case G.D. Raikov obtained earlier corresponding results (under slightly different assumptions) for the Friedrichs extension of the operator.      17. A Phase Field Model for Ductile to Brittle Failure Mode Transition      Martina Hofacker  Christian Miehe 《PAMM》2012,12(1):173-174 The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)      18. 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。      19. 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      20. Distance Labelling Problems for Hypercubes and Hamming Graphs – A Survey      《Electronic Notes in Discrete Mathematics》2007 We survey recent results on a few distance labelling problems for hypercubes and Hamming graphs.

Un Exemple de Transition de Phase

We give a new proof of the non uniqueness of the equilibrium state inF. Hofbauer's example. We extend it to obtain examples with any finite number of ergodic equilibrium states.     

Phase Transition Models in Atmospheric Dynamics

Abstract From both the theoretical and numerical viewpoints, we study a system of differential inclusions describing the evolution of the temperature and the specific humidity distributions in a system of moist air. We allow the so-called saturation concentration parameter to depend on the temperature, and thus we consider more general and interesting phase-change effects than the ones addressed in [2].     

A Phase Transition for Large Values of Bifurcating Autoregressive Models

Journal of Theoretical Probability - We describe the asymptotic behavior of the number $$Z_n[a_n,infty )$$ of individuals with a large value in a stable bifurcating autoregressive process, where...     

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.     

Dependence of the Phase Transition Temperature on the Domain Size

Dependence of the phase transition temperature on the domain size is investigated for a double-well quadratic potential. It is shown that for a domain whose boundary is subjected to a hydrostatical pressure, the temperature of phase transitions is independent of the domain and the surface tension coefficient and depends exclusively on the properties of the elastic media. If the displacement field vanishes on the boundary, then for sufficiently small domains, the temperature also does not depend on the surface tension and domain size and is determined by properties of the elastic media only. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 98–113.     

Multiple Phase Transitions in Long‐Range First‐Passage Percolation on Square Lattices

We consider a model of long‐range first‐passage percolation on the d‐dimensional square lattice ?d in which any two distinct vertices x,y ? ?^d are connected by an edge having exponentially distributed passage time with mean ‖ x – y ‖^α+o(1), where α > 0 is a fixed parameter and ‖·‖ is the l¹–norm on ?^d. We analyze the asymptotic growth rate of the set ß_t, which consists of all x ? ?^d such that the first‐passage time between the origin 0 and x is at most t as t → ∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α < ^d, (ii) stretched exponential growth for α ? d,2d), (iii) superlinear growth for α ? (2d,2d + 1), and finally (iv) linear growth for α > 2d + 1 like the nearest‐neighbor first‐passage percolation model corresponding to α=∞. © 2015 Wiley Periodicals, Inc.     

Phase Transition in the Mechanics of Continuous Media for Big Loading

In this note a matrix partial differential operator is considered. It is shown that under certain conditions it defines a closed operator with nonempty resolvent set, and its essential spectrum is determined. In the symmetric case G.D. Raikov obtained earlier corresponding results (under slightly different assumptions) for the Friedrichs extension of the operator.     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Distance Labelling Problems for Hypercubes and Hamming Graphs – A Survey

We survey recent results on a few distance labelling problems for hypercubes and Hamming graphs.