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Consider the complete graph on n vertices. To each vertex assign an Ising spin that can take the values - 1 or + 1 . Each spin i ∈ [ n ] = { 1 , 2 , ⋯ , n } interacts with a magnetic field h ∈ [ 0 , ∞) , while each pair of spins i , j ∈ [ n ] interact with each other at coupling strength n - 1 J (i) J (j) , where J = (J (i)) i ∈ [ n ] are i.i.d. non-negative random variables drawn from a probability distribution with finite support. Spins flip according to a Metropolis dynamics at inverse temperature β ∈ (0 , ∞) . We show that there are critical thresholds β c and h c (β) such that, in the limit as n → ∞ , the system exhibits metastable behaviour if and only if β ∈ (β c , ∞) and h ∈ [ 0 , h c (β)) . Our main result is a sharp asymptotics, up to a multiplicative error 1 + o n (1) , of the average crossover time from any metastable state to the set of states with lower free energy. We use standard techniques of the potential-theoretic approach to metastability. The leading order term in the asymptotics does not depend on the realisation of J, while the correction terms do. The leading order of the correction term is n times a centred Gaussian random variable with a complicated variance depending on β , h , on the law of J and on the metastable state. The critical thresholds β c and h c (β) depend on the law of J, and so does the number of metastable states. We derive an explicit formula for β c and identify some properties of β ↦ h c (β) . Interestingly, the latter is not necessarily monotone, meaning that the metastable crossover may be re-entrant. [ABSTRACT FROM AUTHOR]

We consider a directed polymer of length N interacting with a linear interface. The monomers carry i.i.d. random charges (ω i) i = 1 N taking values in R with mean zero and variance one. Each monomer i contributes an energy (β ω i - h) φ (S i) to the interaction Hamiltonian, where S i ∈ Z is the height of monomer i with respect to the interface, φ : Z → [ 0 , ∞) is the interaction potential, β ∈ [ 0 , ∞) is the inverse temperature, and h ∈ R is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on Z . We study both the quenched and the annealed free energy per monomer in the limit as N → ∞ . We show that each exhibits a phase transition along a critical curve in the (β , h) -plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as β , h ↓ 0 in three different regimes for the tail exponent of φ . [ABSTRACT FROM AUTHOR]

For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (‘hard constraint’), while the canonical ensemble requires the constraint to be met only on average (‘soft constraint’). It is known that breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-zero specific relative entropy of the two ensembles. In this paper we analyse a formula for the relative entropy of generic discrete random structures recently put forward by Squartini and Garlaschelli. We consider the case of a random graph with a given degree sequence (configuration model), and show that in the dense regime this formula correctly predicts that the specific relative entropy is determined by the scaling of the determinant of the matrix of canonical covariances of the constraints. The formula also correctly predicts that an extra correction term is required in the sparse regime and in the ultra-dense regime. We further show that the different expressions correspond to the degrees in the canonical ensemble being asymptotically Gaussian in the dense regime and asymptotically Poisson in the sparse regime (the latter confirms what we found in earlier work), and the dual degrees in the canonical ensemble being asymptotically Poisson in the ultra-dense regime. In general, we show that the degrees follow a multivariate version of the Poisson-Binomial distribution in the canonical ensemble. [ABSTRACT FROM AUTHOR]

Abstract: We obtain bounds for the expected loss of torsional rigidity of a cylinder C L of length L and planar cross‐section Ω due to a Brownian fracture that starts at a random point in C L and runs until the first time it exits C L. These bounds are expressed in terms of the geometry of the cross‐section Ω ⊂ R 2. It is shown that if Ω is a disc with radius R, then in the limit as L → ∞ the expected loss of torsional rigidity equals c R 5 for some c ∈ ( 0 , ∞ ). We derive bounds for c in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in R 3 with radius 1, and runs until the first time it exits this ball. [ABSTRACT FROM AUTHOR]

In this paper we study a model describing a copolymer in a micro-emulsion. The copolymer consists of a random concatenation of hydrophobic and hydrophilic monomers, the micro-emulsion consists of large blocks of oil and water arranged in a percolation-type fashion. The interaction Hamiltonian assigns energy -α to hydrophobic monomers in oil and energy -β to hydrophilic monomers in water, where α,β are parameters that without loss of generality are taken to lie in the cone {(α,β) ∈ R2: α ≥ |β|}. Depending on the values of these parameters, the copolymer either stays close to the oilwater interface (localization) or wanders off into the oil and/or the water (delocalization). Based on an assumption about the strict concavity of the free energy of a copolymer near a linear interface, we derive a variational formula for the quenched free energy per monomer that is column-based, i.e., captures what the copolymer does in columns of different type. We subsequently transform this into a variational formula that is slope-based, i.e., captures what the polymer does as it travels at different slopes, and we use the latter to identify the phase diagram in the (α,β)-cone. There are two regimes: supercritical (the oil blocks percolate) and subcritical (the oil blocks do not percolate). The supercritical and the subcritical phase diagram each have two localized phases and two delocalized phases, separated by four critical curves meeting at a quadruple critical point. The different phases correspond to the different ways in which the polymer moves through the micro-emulsion. The analysis of the phase diagram is based on three hypotheses about the possible frequencies at which the oil blocks and the water blocks can be visited. We show that these three hypotheses are plausible, but do not provide a proof. [ABSTRACT FROM AUTHOR]

Abstract: We derive a lace expansion for the survival probability for critical spread-out oriented percolation above dimensions, i.e., the probability that the origin is connected to the hyperplane at time n, at the critical threshold . Our lace expansion leads to a non-linear recursion relation for , with coefficients that we bound via diagrammatic estimates. This lace expansion is for point-to-plane connections and differs substantially from previous lace expansions for point-to-point connections. In particular, to be able to deduce the asymptotics of for large n, we need to derive the recursion relation up to quadratic order. The present paper is Part II in a series of two papers. In Part I, we use the recursion relation and the diagrammatic estimates to prove that , and also deduce consequences of this asymptotics for the geometry of large critical clusters and for the incipient infinite cluster. [Copyright &y& Elsevier]

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/ Bn 2 as $$n\to\infty$$ , where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/ Bn as $$n\to\infty$$ . The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider an arbitrary irreducible random walk on , , with i.i.d. increments, together with an arbitrary i.i.d. random scenery. Walk and scenery are assumed to be independent. Random walk in random scenery (RWRS) is the random process where time is indexed by , and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. Bad configurations for RWRS are the discontinuity points of the conditional probability distribution for the configuration at the origin of time given the configuration at all other times. We show that the set of bad configurations is non-empty. We give a complete description of this set and compute its probability under the random scenery measure. Depending on the type of random walk, this probability may be zero or positive. For simple symmetric random walk we get three different types of behavior depending on whether , or . Our classification is actually valid for a class of subshifts having a certain determinative property, which we call specifiable, of which RWRS is an example. We also consider bad configurations w.r.t. a finite time interval (replacing the origin) and obtain an almost complete generalization of our results. Remarkably, this extension turns out to be somewhat delicate. [Copyright &y& Elsevier]

We consider a heteropolymer, consisting of an i.i.d. concatenation of hydrophilic and hydrophobic monomers, in the presence of water and oil arranged in alternating layers. The heteropolymer is modelled by a directed path ($(i,\; S\_i)\_\{i\backslash in\; \{\backslash Bbb\; N\}\_0\}$, where the vertical component lives on $\{\backslash Bbb\; Z\}$, and the layers are horizontal with equal width. The path measure for the vertical component is given by that of simple random walk multiplied by an exponential weight factor that favors matches and disfavors mismatches between the monomers and the medium. We study the vertical motion of the heteropolymer as a function of its total length n when the width of the layers is dn and the parameters in the exponential weight factor are such that the heteropolymer tends to stay close to an interface (“localized regime”). In the limit as n→∞ and under the condition that limn→∞dn/log log n=∞ and limn→∞dn/log n=0, we show that the vertical motion is a diffusive hopping between neighboring interfaces on a time scale exp[χdn(1+o(1))], where χ is computed explicitly in terms of a variational problem. An analysis of this variational problem sheds light on the optimal hopping strategy. [ABSTRACT FROM AUTHOR]

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on Z[sup d] where loops of length m are penalised by a factor e[sup -beta/m[sup p]] (0 < beta much less than 1) when: (1) d > 4, p is greater than or equal to 0; (2) d less than or equal to 4, p > 4 - d/2. In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d > 4, p = 0. In addition, we prove a local central limit theorem, with the exception of the case d > 4, p = 0. [ABSTRACT FROM AUTHOR]

In this paper, which is a culmination of our previous research efforts, we provide a general framework for studying mixing profiles of non-backtracking random walks on dynamic random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph tends to infinity. Subject to mild general conditions, we link two mixing times: one for a static version of the random graph, the other for a class of dynamic versions of the random graph in which the edges are randomly rewired but the degrees are preserved. With the help of coupling arguments we show that the link is provided by the probability that the random walk has not yet stepped along a previously rewired edge. To demonstrate the utility of our framework, we rederive our earlier results on mixing profiles for global edge rewiring under weaker assumptions, and extend these results to an entire class of rewiring dynamics parametrised by the range of the rewiring relative to the position of the random walk. Along the way we establish that all the graph dynamics in this class exhibit the trichotomy we found earlier, namely, no cut-off, one-sided cut-off or two-sided cut-off. For interpolations between global edge rewiring, the only Markovian graph dynamics considered here, and local edge rewiring (i.e., only those edges that are incident to the random walk can be rewired), we show that the trichotomy splits further into a hexachotomy, namely, three different mixing profiles with no cut-off, two with one-sided cutoff, and one with two-sided cut-off. Proofs are built on a new and flexible coupling scheme, in combination with sharp estimates on the degrees encountered by the random walk in the static and the dynamic version of the random graph. Some of these estimates require sharp control on possible short-cuts in the graph between the edges that are traversed by the random walk. • We give a framework to study mixing of non-backtracking random walks on dynamic graphs. • We describe mixing profiles for a class of graph dynamics parametrised by their range. • For a class of graph dynamics we observe a hexachotomy in their mixing profiles. [ABSTRACT FROM AUTHOR]

It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by longrange interactions or nonadditivity, (2) mathematically, nonequivalence is determined by a different large-deviation behavior of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in gener [ABSTRACT FROM AUTHOR]

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established. [ABSTRACT FROM AUTHOR]

Focuses on efforts to archive web sites of political parties in the Netherlands. Appraisal of some international initiatives to archive such web sites; Details of the Dutch project, Archipol, to archive web sites of political parties; Significance of archived web sites to political scientists; Scope of further research in the area.