Glauber Dynamics

We construct a Glauber dynamics on {0, 1}ℛ, ℛ a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.

We study local Markov chains for sampling 3-colorings of the discrete torus T L,d = {0, . . . , L -1} d . We show that there is a constant ρ ≈ .22 such that for all even L ≥ 4 and d sufficiently large, certain local Markov chains require exponential time to converge to equilibrium. More precisely, if M is a Markov chain on the set of proper 3-colorings of T L,d that updates the color of at most ρL d vertices at each step and whose stationary distribution is uniform, then the convergence to stationarity of M is exponential in L d-1 . Our proof is based on a conductance argument that builds on sensitive new combinatorial enumeration techniques.

In this work we present a new method to calculate the classical magnetic properties of singledomain nanoparticles. Based on the Bethe-Peierls (pair) approximation, we developed a simple system of equations for the classical magnetization of spins at any position within the nanoparticle. Nearest neighbor pair correlations are treated exactly for Ising spins, and the method can be generalized for various lattice symmetries. The master equation is solved for the Glauber dynamics (single-spin-flip) in order to obtain the time evolution of the magnetization. The capabilities of the model are demonstrated through nontrivial calculations of hysteresis loops as well as field cooling (FC) and zero field cooling (ZFC) magnetization curves of heterogeneous non-interacting nanoparticles. The present method automatically incorporates the temperature and could be adapted to describe an ensemble of interacting nanoparticles.

We consider, as in part I (see above), a random walk X t ∈ℤ ν , t∈ℤ + , and a dynamical random field ξ t (x), x∈ℤ ν , in mutual interaction with each other. The model is a perturbation of an unperturbed model in which walk and field evolve independently. Here we consider the environment process in a frame of reference that moves with the walk, i.e., the “field from the point of view of the particle” η t (·)=ξ t (X t +·). We prove that its distribution tends, as t→∞, to a limiting distribution μ, which is absolutely continuous with respect to the unperturbed equilibrium distribution. We also prove that, for ν≥3, the time correlations of the field η t decay as const·e -αt /t ν/2 .

We consider a two-type stochastic competition model on the integer lattice Z d . The model describes the space evolution of two "species" competing for territory along their boundaries. Each site of the space may contain only one representative (also referred to as a particle) of either type. The spread mechanism for both species is the same: each particle produces offspring independently of other particles and can place them only at the neighboring sites that are either unoccupied, or occupied by particles of the opposite type. In the second case, the old particle is killed by the newborn. The rate of birth for each particle is equal to the number of neighboring sites available for expansion. The main problem we address concerns the possibility of the long-term coexistence of the two species. We have shown that if we start the process with finitely many representatives of each type, then, under the assumption that the limit set in the corresponding first passage percolation model is uniformly curved, there is positive probability of coexistence.

We introduce a novel measure, Fisher transfer entropy (FTE), which quantifies a gain in sensitivity to a control parameter of a state transition, in the context of another observable source. The new measure captures both transient and contextual qualities of transfer entropy and the sensitivity characteristics of Fisher information. FTE is exemplified for a ferromagnetic two-dimensional lattice Ising model with Glauber dynamics and is shown to diverge at the critical point.

Hist.oricament.e, a conexão ent.re invariãncia de escala e invariância conforme é conhecida desde o io1cio do século pelos fi sicos t.eóricos. Por exemplo, as equaçeses de Maxwell no vácuo No ent.ant.o, soment.e em 1970 com são invariant.es t.ant.o por t.ransformaçeses conformes 1 t.ransformaçeses de escala. como por Polyakov2 surgiu o .primeiro t.rabalho referent.e à aplicação de invariância conforme a fenÔmenos cri t.icos. Nest.a época as at.ençeses est.avam volt.adas aos princi pios de invariância de escala e K. Wilson:;!criou uma t.écnica <grupo de renormalização) bast.ant.e eficaz no cálculo dos expoent.es cri t.icos dos modelos de mecânica est.at.i st.ica. Mais recent.ement.e em um t.rabalho realizado por Belavin, Polyakov e Zamolodchikov4(BPZ) as idéias de invariância conforme revelaram-se de :;rande valia no est.udo de fenomenos cri t.icos. Part..icularment.e em duas dimenseses o grupo conforme é isomórfico ao ~rupo de t-unções anall t..icas t..endo port.ant..o dimensão in:ftnit..a. A ál~ebra correspondent.e a est..e ~rupo, chamada ál~ebra de Virasoro. já havia sido est..udada no cont..ext..o de t..eoria de cordas. Subsequent.ement..e o mat..emát..icoV. Kac e colaboradores!:> est.udaram a represent..ação dest..a ál~ebra. Em seu t..rabalho BPZ most.raram que cada operador de escala dos sist..emas bi-dimensionais no pont.o crít..ico corresponde a uma represent..ação dest..a ál~ebra, indexada pelo valor da sua car~a cent..ral ou anomalia conforme c. 2 Se est..a SERViÇO DE sTii"LIOTE CÃ-"E INtõRi~AO _ IFOSC FISIC A represent.ação Íor de uma espécie part.icularment.e simples, correspondendo ao anulament.o de uma cert.a quant.idade chamada det.erminant.e de Kac, ent.ão não soment.e os expoent.es cri t.icos, mas t.odas as Íunções de correlação de muit.os pont.os no pont.o cri t.ico podem ser obt.idas; nest.a linha seguiram os t.rabalhos de Dot.senko6 e Dot.senko e Fat.eev 7. Nisso t.udo, a suposição Íeit.a por BPZ que soment.e. represent.ações degeneradas apareceriam, ainda não est.ava clara, quando Friedan, Qiu e Shenker8 most.raram que se a t.eoria é unit.ária e que o valor da anomalia conÍorme c sat.isÍaz c<1, de Íat.o t.ais represent.ações são as únicas possiveis. Independent.ement.e, nest.a mesma época, já com maiores Íacilidades na comput.ação númerica, a t.eoria de escala para sint.emas Íini t.os t.ornou-se uma Íerrament.a import.ant.e no est.udo de t.ransições de Íase (Barber9). Cardy10 ut.ilizando-se das idéias de invariância conÍorme acima descrit.as most.rou ser possi vel mapear uma geomet.ria em out.ra mais simples, podendo dest.a maneira pode-se ext.rair propriedades de sist.emas inÍini t.os a part.ir de sist.emas Íini t.os. o sucesso dos result.ados numéricos obt.idos dest.a f"orma f"irmaram de maneira convincent.e o pri ncipio de invariância conÍorme nos Íenómenos cri t.icos. , Est.e t.rabalho cont.ém os desenvolviment.os da t.eoria conforme no capi t.ulo (2) e da t.eoria de escala para sist.emas Íinit.os no capi t.ulo (3). Hais adiant.e nos capi t.ulos (4) e (5) são apresent.ados as aplicações e result.ados das t.écnicas desenvolvidas em dois modelos, e Íinalment.e no capi t.ulo (6) as discussões e FIGURA (lI-l) FIGURA (lI-2) Uma trans.tormação de escala por dilatação no espaço É? ilustrado no figura (11-D. Na (11-2) temos uma transformação conforme.

We derive upper and lower bounds for the spectral gap of the random energy model under Metropolis dynamics which are sharp in exponential order. They are based on the variational characterization of the gap. For the lower bound, a Poincaré inequality derived by Diaconis and Stroock is used. The scaled asymptotic expression is a linear function of the temperature. The corresponding function for a global version of the dynamics exhibits phase transition instead. We also study the dependence of lower order terms on the volume. In the global dynamics, we observe a phase transition. For the local dynamics, the expressions we have, which are possibly not sharp, do not change their order of dependence on the volume as the temperature changes.

We analyse the lower non trivial part of the spectrum of the generator of the Glauber dynamics for a d-dimensional nearest neighbour Ising model with a bounded random potential. We prove conjecture 1 in [AMSZ]: for sufficently large values of the temperature, the first band of the spectrum of the generator of the process coincides with a closed non random segment of the real line.

We study the Glauber dynamics Markov chain for k-colourings of trees with maximum degree ∆. For k ≥ 3, we show that the mixing time on the complete tree is n θ(1+∆/(k log ∆)). For k ≥ 4 we extend our analysis to show that the mixing time on any tree is at most n O(1+∆/(k log ∆)). Our proof uses a weighted canonical paths analysis and introduces a variation of the block dynamics that exploits the differing relaxation times of blocks.

Recent results have shown that the Glauber dynamics for graph colourings has optimal mixing time when (i) the graph is triangle-free and D-regular and the number of colours k is a small constant fraction smaller than 2D, or (ii) the graph has maximum degree D and k = 2D. We extend both these results to prove that the Glauber dynamics has optimal mixing time when the graph has maximum degree D and the number of colours is a small constant fraction smaller than 2D.

We consider Markov random fields of discrete spins on the lattice Z d . We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is "sufficiently" subcritical, then we prove the Poincaré inequality. In the whole subcritical regime, we have a weak Poincaré inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.

We consider a one-dimensional version of the model introduced in Ref, 1. At each site of Z there is a particle with spin + 1. Particles move according to the Stirring Process and spins change according to the Glauber dynamics. In the hydrodynamical limit, with the stirring process suitably speeded up, the local magnetic density m,(r) is proven in Ref. 1 to satisfy the reaction-diffusion equation V(m)=-89 4, ~ and fl>0, c~ and fl being determined by the parameters of the Glauber dynamics. In the present paper we consider an initial state with zero magnetization, mo(r)=O. We then prove that at long times, before taking the hydrodynamical limit, the evolution departs from that predicted by (*) and that the microscopic state becomes a nontrivial mixture of states with different magnetizations.

The evolutions of states is described corresponding to the Glauber dynamics of an infinite system of interacting particles in continuum. The description is conducted on both micro-and mesoscopic levels. The microscopic description is based on solving linear equations for correlation functions by means of an Ovsjannikov-type technique, which yields the evolution in a scale of Banach spaces. The mesoscopic description is performed by means of the Vlasov scaling, which yields a linear infinite chain of equations obtained from those for the correlation function. Its main peculiarity is that, for the initial correlation function of the inhomogeneous Poisson measure, the solution is the correlation function of such a measure with density which solves a nonlinear differential equation of convolution type.

In this paper we study homogeneous Gibbs measures on a Cayley tree, subjected to an infinite-temperature Glauber evolution, and consider their (non-)Gibbsian properties. We show that the intermediate Gibbs state (which in zero field is the free-boundary-condition Gibbs state) behaves different from the plus and the minus state. E.g. at large times, all configurations are bad for the intermediate state, whereas the plus configuration never is bad for the plus state. Moreover, we show that for each state there are two transitions. For the intermediate state there is a transition from a Gibbsian regime to a non-Gibbsian regime where some, but not all configurations are bad, and a second one to a regime where all configurations are bad. For the plus and minus state, the two transitions are from a Gibbsian regime to a non-Gibbsian one and then back to a Gibbsian regime again.

Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which does not involve the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transitiom. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph.

We consider spin systems with nearest‐neighbor interactions on an n‐vertex d‐dimensional cube of the integer lattice graph . We study the effects that the strong spatial mixing condition (SSM) has on the rate of convergence to equilibrium of nonlocal Markov chains. We prove that when SSM holds, the relaxation time (i.e., the inverse spectral gap) of general block dynamics is O(r), where r is the number of blocks. As a second application of our technology, it is established that SSM implies an O(1) bound for the relaxation time of the Swendsen‐Wang dynamics for the ferromagnetic Ising and Potts models. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is . Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra.

Understanding the strategic behavior of miners in a blockchain is of great importance for its proper operation. A common model for mining games considers an infinite time horizon, with players optimizing asymptotic average objectives. Implicitly, this assumes that the asymptotic behaviors are realized at human-scale times, otherwise invalidating current models. We study the mining game utilizing Markov Decision Processes. Our approach allows us to describe the asymptotic behavior of the game in terms of the stationary distribution of the induced Markov chain. We focus on a model with two players under immediate release, assuming two different objectives: the (asymptotic) average reward per turn and the (asymptotic) percentage of obtained blocks. Using tools from Markov chain analysis, we show the existence of a strategy achieving slow mixing times, exponential in the policy parameters. This result emphasizes the imperative need to understand convergence rates in mining games, validating the standard models. Towards this end, we provide upper bounds for the mixing time of certain meaningful classes of strategies. This result yields criteria for establishing that long-term averaged functions are coherent as payoff functions. Moreover, by studying hitting times, we provide a criterion to validate the common simplification of considering finite states models. For both considered objectives functions, we provide explicit formulae depending on the stationary distribution of the underlying Markov chain. In particular, this shows that both mentioned objectives are not equivalent. Finally, we perform a market share case study in a particular regime of the game. More precisely, we show that an strategic player with a sufficiently large processing power can impose negative revenue on honest players.

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system size N. For the largest of these barriers we find a behavior proportional to N ~ with 2 = 1/3.

We consider pairwise Markov random fields which have a number of important applications in statistical physics, image processing and machine learning such as Ising model and labeling problem to name a couple. Our own motivation comes from the need to produce synthetic models for social networks with attributes. First, we give conditions for rapid mixing of the associated Glauber dynamics and consider interesting particular cases. Then, for pairwise Markov random fields with submodular energy functions we construct monotone perfect simulation.

CORE View metadata, citation and similar papers at core.ac.uk provided by Repositório Institucional da UFSC v vi "Eu não adivinho. Como cientista eu chego a conclusões baseadas em observação e experimentação." Dr. Sheldon Cooper vii viii RESUMO Este trabalho divide-se em duas partes. Na primeira, nós calculamos o expoente dinâmico do algoritmo de Niedermayer aplicado aos modelos de Ising e XY em duas dimensões, para vários valores do parâmetro E 0 (o qual, resumidamente, controla o tamanho médio das ilhas a serem modificadas). Para E 0 = −1 nós reobtemos o algoritmo de Metropolis e para E 0 = 1 reobtemos o algoritmo de Wolff. Para −1 < E 0 < 1, nós mostramos que o tamanho médio das ilhas inicialmente cresce com o tamanho linear do sistema, L, mas eventualmente satura em um determinado tamanhoL, que depende de E 0. Para L >L o algoritmo de Niedermayer se comporta como o algoritmo de Metropolis, istoé, tem o mesmo expoente dinâmico. Para E 0 > 1, os tempos de auto-correlação são sempre maiores que para E 0 = 1 (Wolff) e, mais importante, sempre crescem mais rápido que uma lei de potência de L. Portanto, mostramos que a melhor escolha do parâmetro E 0é o que retoma o algoritmo de Wolff. Nós também obtemos o comportamento dinâmico do algoritmo de Wolff; apesar de não conclusivo, propusemos uma lei de escala para o tempo de auto-correlação. Na segunda parte nós estudamos o modelo de Potts numa rede retangular com modulações aperiódicas nas interações ao longo de uma direção. Os resultados numéricos foram obtidos utilizando o algoritmo de Wolff para diferentes tamanhos de redes, permitindo que o método de escala de tamanho finito fosse utilizado. Foram utilizadas 3 sequências aperiódicas autoduais, as quais permitem resultados mais precisos, uma vez que a temperatura crítica pode ser conhecida exatamente. Nós analisamos 3 modelos, com seis, oito e quinze estados, todos com transições de primeira ordem no sistema uniforme. Mostramos que o critério de Harris-Luck, originalmente introduzido para o estudo de transições contínuas,é obedecido também para transições de primeira ordem. Nossos resultados indicam que a nova classe de universalidadé e dependente do número de estados do modelo de Potts. Como esperado, observamos uma dependência log-periódica da magnetização e da susceptibilidade com o tamanho do sistema finito.

We study a two-dimensional nonequilibrium Ashkin-Teller model based on competing dynamics induced by contact with a heat bath at temperature T, and subject to an external source of energy. The dynamics of the system is simulated by two competing stochastic processes: a Glauber dynamics with probability p, which simulates the contact with the heat bath; and a Kawasaki dynamics with probability 1Ϫp, which takes into account the flux of energy into the system. Monte Carlo simulations were employed to determine the phase diagram for the stationary states of the model and the corresponding critical exponents. The phase diagrams of the model exhibit a self-organization phenomenon for certain values of the fourth coupling interaction strength. On the other hand, from exponent calculations, the equilibrium critical behavior is preserved when nonequilibrium conditions are applied.

An open ferromagnetic Ashkin-Teller model with spin variables 0, ±1 is studied by standard Monte Carlo simulations on a square lattice in the presence of competing Glauber and Kawasaki dynamics. The Kawasaki dynamics simulates spin-exchange processes that continuously flow energy into the system from an external source. Our calculations reveal the presence, in the model, of tricritical points where first order and second order transition lines meet. Beyond that, several self-organized phases are detected when Kawasaki dynamics become dominant. Phase diagrams that comprise phase boundaries and stationary states have been determined in the model parameters’ space. In the case where spin-phonon interactions are incorporated in the model Hamiltonian, numerical results indicate that the paramagnetic phase is stabilized and almost all of the self-organized phases are destroyed.

After World War II, quite a few mathematicians were attracted to the modeling of phase transitions as this area of physics was seeing considerable mathematical difficulties. This paper studies their contributions to the physics of phase transitions, and in particular those of the by far most productive and successful of them, the Polish-American mathematician Mark Kac (1914-1984). The focus is on the resources, values, and traditions that the mathematicians brought with them and how these differed from those of contemporary physicists as well as the mathematicians' relations with the physicists in terms of collaboration and reception of results. 1. Introduction After World War II, quite a few mathematicians, including Mark Kac, John von Neumann, and Nobert Wiener, worked on the physical problem of phase transitions, i.e. changes in the state of matter caused by gradual changes of physical parameters such as the condensation of a gas to a liquid and the loss of magnetization of a ferromagnet above a certain temperature (called the Curie temperature). The significance of these mathematicians was not so much that they brought mathematical rigor to the theoretical description of the phenomena, 1 but that they applied their mathematical

In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.

In this paper we analyze several anisotropic bootstrap percolation models in three dimensions. We present the order of magnitude for the metastability thresholds for a fairly general class of models. In our proofs, we use an adaptation of the technique of dimensional reduction. We find that the order of the metastability threshold is generally determined by the 'easiest growth direction' in the model. In contrast to anisotropic bootstrap percolation in two dimensions, in three dimensions the order of the metastability threshold for anisotropic bootstrap percolation can be equal to that of isotropic bootstrap percolation.

Each oscillator in a linear chain (a string) interacts with a local Ising spin in contact with a thermal bath. These spins evolve according to Glauber dynamics. Below a critical temperature, there appears an equilibrium, time-independent, rippled state in the string that is accompanied by a nonzero spin polarization. On the other hand, the system is shown to form "metastable", nonequilibrium but long-lived ripples in the string for slow spin relaxation. The system vibrates rapidly about these quasi-stationary states which can be described as snapshots of a coarse-grained stroboscopic map. For moderate observation times, ripples are observed irrespective of the final thermodynamically stable state (rippled or not). Interestingly, the system can be considered as a "minimal" model to understand rippling in clamped graphene sheets.

The paper examines an infinitely repeated 3-player extension of the Prisoner's Dilemma game. We consider a 3-player game in the normal form with incomplete information, in which each player has two actions. We assume that the game is symmetric and repeated infinitely many times. At each stage, players make their choices knowing only the average payoffs from previous stages of all the players. A strategy of a player in the repeated game is a function defined on the convex hull of the set of payoffs. Our aim is to construct a strong Nash equilibrium in the repeated game, i.e. a strategy profile being resistant to deviations by coalitions. Constructed equilibrium strategies are safe, i.e. the non-deviating player payoff is not smaller than the equilibrium payoff in the stage game, and deviating players' payoffs do not exceed the non-deviating player payoff more than by a positive constant which can be arbitrary small and chosen by the non-deviating player. Our construction is inspired by Smale's good strategies described in [17], where the repeated Prisoner's Dilemma was considered. In proofs we use arguments based on approachability and strong approachability type results.

The order-disorder phase transition in Ising spin systems is investigated, with flows between states. The rates of these flows depend on the states of a local environment of a spin. This dependence plays the role of the spinspin interaction. The number of rate parameters depend on the system topology and is lowered by the up-down symmetry. Numerical calculations are supplemented by mean field approach. The results can be useful in modelling social systems, where temperature is not defined.

We provide an optimally mixing Markov chain for 6-colorings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colorings has strong spatial mixing. Four and five are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colorings of the square grid.

Let (X, d) be a locally compact separable ultra-metric space. Given a reference measure μ on X and a step length distribution σ on [0 , ∞), we construct a symmetric Markov semigroup {P t}t≥0 acting in L(X,μ). Let {Xt} be the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator L and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting our results are new. We also elaborate the relation between our processes and Kigami’s jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we explain the fractional derivative on the p-adic integers and the corresponding random walk on the associated tree.

A lattice gas with non-conserved spin flip dynamics (of both non-Glauber and Glauber types) is considered at T°T c , the critical temperature. For arbitrary supersaturation, S, a general expression for the inverse of the nucleation rate along the lowest energy path is derived. The exponential part is identical to the one by Neves and Schonmann [Commun. Math. Phys. 137:20 (1991)]. The preexponential can be expressed in terms of elliptic theta-functions for small S, and in the limits, respectively, of S ± T/f or S°T/f (− f being the nearestneighbor interaction energy), elementary versions of the general expression are further obtained. The preexponential has a smooth component, as well as small-scale modulations which are approximately periodic in the inverse supersaturation. For S°T/f, the smooth part is proportional to`S, in contrast to the zero-T limit where it is linear in S. The latter limit becomes apparent only at extremely low temperatures which are cubic in S.

We consider the problem of generating a random q-colouring of a graph G=(V,E). We consider the simple Glauber Dynamics chain. We show that if for all v ∈ V the average degree of the subgraph H_v induced by the neighbours of v ∈ V is #x226a Δ where Δ is the maximum degree and Δ &gt;c_1\ln n then for sufficiently large c_1, this chain mixes rapidly provided q/Δ &gt;α , where α #x2248 1.763 is the root of α = e^\1/α \. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs G_\n,p\ with p #x226a 1, this beats the 11Δ /6 bound of Vigoda for general graphs.

Let ξ 1 and ξ 2 be two solutions of two stochastic differential equations with respect to Lévy noise taking values in a certain type of Banach space. Let Q 1 and Q 2 be the probability measures on the corresponding Skorohod space induced by ξ 1 and ξ 2 , respectively. In the paper we are interested under which conditions Q 1 is absolute continuous with respect to Q 2. Moreover, we give an explicit formula for the Radon Nikodym derivative of Q 1 with respect to Q 2 .

In this paper we introduce a variant of the honeycomb lattice in which we create defects by randomly exchanging adjacent bonds, producing a random tiling with a distribution of polygon edges. We study the percolation properties on these lattices as a function of the number of exchanged bonds using a novel computational method. We find the site and bond percolation thresholds are consistent with other three-coordinated lattices with the same standard deviation in the degree distribution of the dual; here we can produce a continuum of lattices with a range of standard deviations in the distribution. These lattices should be useful for modeling other properties of random systems as well as percolation.

We have studied the magnetic relaxation behavior of a two-dimensional Ising ferromagnet by Monte Carlo simulation. Our primary goal is to investigate the effects of the system’s geometry (area preserving), boundary conditions and dynamical rules on the relaxation behavior. The Glauber and Metropolis dynamical rules have been employed. The systems with periodic and open boundary conditions are studied. The major findings are the exponential relaxation and the dependence of relaxation time ([Formula: see text]) on the aspect ratio [Formula: see text] (length over breadth having fixed area). A power law dependence ([Formula: see text]) has been observed for larger values of aspect ratio ([Formula: see text]). The exponent ([Formula: see text]) has been found to depend linearly ([Formula: see text]) on the system’s temperature ([Formula: see text]). The transient behaviors of the spin-flip density have been investigated for both surface and bulk/core. The size dependencies of saturated ...

An n-tournament T with vertex set V is simple if there is no subset M of V such that 2 ≤ |M | ≤ n − 2 and for every x ∈ V \ M , either M → x or x → M . The arrow simplicity of a tournament T is the minimal number s(T ) of arcs whose reversal yields a simple tournament. Müller and Pelant proved that s(T ) ≤ n−1 2 , and that equality holds if and only if n ≡ 3 (mod 4) and T is doubly regular. In this paper, we give a refinement of this bound for n 6≡ 3 (mod 4).

The problem of predictability, or “nature vs. nurture”, in both ordered and disordered Ising systems following a deep quench from infinite to zero temperature is reviewed. Two questions are addressed. The first deals with the nature of the final state: for an infinite system, does every spin flip infinitely often, or does every spin flip only finitely many times, or do some spins flip infinitely often and others finitely often? Once this question is determined, the evolution of the system from its initial state can be studied with attention to the issue of how much information contained in the final state depends on that contained in the initial state, and how much depends on the detailed history of the system. This problem has been addressed both analytically and numerically in several papers, and their main methods, results, and conclusions will be reviewed. The discussion closes with some open problems that remain to be addressed.

A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two-or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable L p (L 2)-space with p ∈ (1, 2). This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation.

In this note we discuss metastability in a long-but-finite range disordered model for the glass transition. We show that relaxation is dominated by configuration belonging to metastable states and associate an in principle computable free-energy barrier to the equilibrium relaxation time. Adam-Gibbs-like relaxation times appear naturally in this approach.

For the characterization of multipliers of Lp(Rd) or more generally, of Lp(G) for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra Ap(G) plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of Ap(G). In fact, we demonstrate that it coincides with the space of multipliers of Lp(G),∥·∥p. Given a multiplier T of (Ap(G),∥·∥Ap(G)) and using the embedding (Ap(G),∥·∥Ap(G))↪C0(G),∥·∥∞, the linear functional f↦[T(f)(0)] is bounded, and T can be written as a moving average for some element in the dual PMp(G) of (Ap(G),∥·∥Ap(G)). A key step for this identification is another elementary fact: showing that the multipliers from Lp(G),∥·∥p to C0(G),∥·∥∞ are exactly the convolution operators with kernels in Lq(G),∥·∥q for 1

We consider the metastable behavior of a superposition of a ferromagnetic spin system with a Glauber dynamics and stirring dynamics. Starting from configuration-1, minus spins at all lattice sites in a fixed volume under periodic boundary conditions, the process stays close to this configuration for an unpredictable time until the formation of a droplet, of spins + 1, with a certain critical size and decays to configuration + I in a relatively short time. We observe that the size of the droplet depends on the rate of exclusion.

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.

We consider the complexity of random ferromagnetic landscapes on the hypercube {±1} N given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph G(N, p). Previous results had shown that, with high probability as N → ∞, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized). With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as N → ∞. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.