Convergence of block spins defined by a random field

We study the asymptotic behavior of families of dependent random variables called block spins, which are associated with random fields arising in statistical mechanics. We give sufficient conditions for these families to converge weakly to products of independent Gaussian random variables. We also estimate the error terms involved. In addition we give some conditions which imply that the block spins can converge weakly only to families of normal or degenerate random variables. Central to our proofs is a mixing property which is weaker than strong mixing and which holds for many random fields studied in statistical mechanics. Finally we give a simple method for determining when a stationary random field does not satisfy a strong mixing property. This method implies that the two-dimensional Ising model at the critical temperature is not strong mixing, a result obtained by a different method by M. Cassandro and G. Jona-Lasinio. The method also shows that a stationary, mean-zero, positively correlated Gaussian process indexed by is not strong mixing if its covariance function decreases liket ^– , 0 < < 1.     

The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.     

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.     

Correlated Fractal Percolation and the Palis Conjecture

Let F ₁ and F ₂ be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dim?_H(F ₁) and dim?_H(F ₂). Consider the following question: does dim?_H(F ₁)+dim?_H(F ₂)>1 imply that their algebraic difference F ₁?F ₂ will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.     

Compositions,Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes N _α (t), N _β (t), t>0, we have that \(N_{\alpha}(N_{\beta}(t)) \stackrel{\mathrm{d}}{=} \sum_{j=1}^{N_{\beta}(t)} X_{j}\), where the X _j s are Poisson random variables. We present a series of similar cases, where the outer process is Poisson with different inner processes. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form \(N_{\alpha}(\tau_{k}^{\nu})\), ν∈(0,1], where \(\tau_{k}^{\nu}\) is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form Θ(N(t)), t>0, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.     

Locally Perturbed Random Walks with Unbounded Jumps

Szász and Telcs (J. Stat. Phys. 26(3), 1981) have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if d≥2. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of Szász and Telcs (J. Stat. Phys. 26(3), 1981) to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on Z ^d (d≥2) having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive \(\sqrt{n\log n}\) scaling.     

Corrected Mean-Field Model for Random Sequential Adsorption on Random Geometric Graphs

A notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non-overlapping congruent spheres in the d-dimensional Euclidean space with \(d\ge 2\). Spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres. Due to spatial correlations, characterizing the fraction of accepted spheres remains largely intractable. We study this fraction by taking a novel approach that compares random sequential adsorption in Euclidean space to the nearest-neighbor blocking on a sequence of clustered random graphs. This random network model can be thought of as a corrected mean-field model for the interaction graph between the attempted spheres. Using functional limit theorems, we characterize the fraction of accepted spheres and its fluctuations.     

Noteworthy fractal features and transport properties of Cantor tartans

This Letter is focused on the impact of fractal topology on the transport processes governed by different kinds of random walks on Cantor tartans. We establish that the spectral dimension of the infinitely ramified Cantor tartan $d_s$ is equal to its fractal (self-similarity) dimension D. Consequently, the random walk on the Cantor tartan leads to a normal diffusion. On the other hand, the fractal geometry of Cantor tartans allows for a natural definition of power-law distributions of the waiting times and step lengths of random walkers. These distributions are Lévy stable if $D>1.5$. Accordingly, we found that the random walk with rests leads to sub-diffusion, whereas the Lévy walk leads to ballistic diffusion. The Lévy walk with rests leads to super-diffusion, if $D>\sqrt{3}$, or sub-diffusion, if $1.5<D<\sqrt{3}$.     

Concentration of Measure for Quantum States with a Fixed Expectation Value

Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.     

Subdiffusivity of a Random Walk Among a Poisson System of Moving Traps on $\mathbb {Z}$

We consider a random walk among a Poisson system of moving traps on \(\mathbb {Z}\). In earlier work (Drewitz et al. Springer Proc. Math. 11, 119-158 2012), the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time t in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk’s range.     

Pinning Model in Random Correlated Environment: Appearance of an Infinite Disorder Regime

We study the influence of a correlated disorder on the localization phase transition in the pinning model (Random polymer models, 2007). When correlations are strong enough, an infinite disorder regime arises: large and frequent attractive regions appear in the environment. We present here a pinning model in random binary ( $\{-1,1\}$ -valued) environment. Defining infinite disorder via the requirement that the probability of the occurrence of a large attractive region is sub-exponential in its size, we prove that it coincides with the fact that the critical point is equal to its minimal possible value, namely $h_c(\beta )=-\beta $ . We also stress that in the infinite disorder regime, the phase transition is smoother than in the homogeneous case, whatever the critical exponent of the homogeneous model is: disorder is therefore always relevant. We illustrate these results with the example of an environment based on the sign of a Gaussian correlated sequence, in which we show that the phase transition is of infinite order in presence of infinite disorder. Our results contrast with results known in the literature, in particular in the case of an IID disorder, where the question of the influence of disorder on the critical properties is answered via the so-called Harris criterion, and where a conventional relevance/irrelevance picture holds.     

Free Energy of the Cauchy Directed Polymer Model at High Temperature

We study the Cauchy directed polymer model on \(\mathbb {Z}^{1+1}\), where the underlying random walk is in the domain of attraction to the 1-stable law. We show that, if the random walk satisfies certain regularity assumptions and its symmetrized version is recurrent, then the free energy is strictly negative at any inverse temperature \(\beta >0\). Moreover, under additional regularity assumptions on the random walk, we can identify the sharp asymptotics of the free energy in the high temperature limit, namely,

$$\begin{aligned} \lim \limits _{\beta \rightarrow 0}\beta ^{2}\log (-p(\beta ))=-c. \end{aligned}$$

     

Global Solutions of the Boltzmann Equation Over {\mathbb{R}^D} Near Global Maxwellians with Small Mass

We consider the random regular k-nae- sat problem with n variables, each appearing in exactly d clauses. For all k exceeding an absolute constant \({{\it k}_0}\), we establish explicitly the satisfiability threshold \({{{d_\star} \equiv {d_\star(k)}}}\). We prove that for \({{d < d_\star}}\) the problem is satisfiable with high probability, while for \({{d > d_\star}}\) the problem is unsatisfiable with high probability. If the threshold \({{d_\star}}\) lands exactly on an integer, we show that the problem is satisfiable with probability bounded away from both zero and one. This is the first result to locate the exact satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon identified by Krz?aka?a et al. [Proc Natl Acad Sci 104(25):10318–10323, 2007]. Our proof verifies the one-step replica symmetry breaking formalism for this model. We expect our methods to be applicable to a broad range of random constraint satisfaction problems and combinatorial problems on random graphs.     

A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one ifd>2 and <₀, where ₀ is defined in terms of the probability that the symmetric nearest neighbor random walk on thed-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.     

Random Walk with Barycentric Self-interaction

We study the asymptotic behaviour of a d-dimensional self-interacting random walk (X _n ) _n∈? (?:={1,2,3,…}) which is repelled or attracted by the centre of mass \(G_{n} = n^{-1} \sum_{i=1}^{n} X_{i}\) of its previous trajectory. The walk’s trajectory (X ₁,…,X _n ) models a random polymer chain in either poor or good solvent. In addition to some natural regularity conditions, we assume that the walk has one-step mean drift

$\mathbb{E}[X_{n+1} - X_n \mid X_n - G_n = \mathbf{x}] \approx\rho\|\mathbf{x}\|^{-\beta}\hat{ \mathbf{x}}$

for ρ∈? and β≥0. When β<1 and ρ>0, we show that X _n is transient with a limiting (random) direction and satisfies a super-diffusive law of large numbers: n ^?1/(1+β) X _n converges almost surely to some random vector. When β∈(0,1) there is sub-ballistic rate of escape. When β≥0 and ρ∈? we give almost-sure bounds on the norms ‖X _n ‖, which in the context of the polymer model reveal extended and collapsed phases.

Analysis of the random walk, and in particular of X _n ?G _n , leads to the study of real-valued time-inhomogeneous non-Markov processes (Z _n ) _n∈? on [0,∞) with mean drifts of the form

$ \mathbb{E}[ Z_{n+1} - Z_n \mid Z_n = x ] \approx\rho x^{-\beta} - \frac {x}{n},$

(0.1)

where β≥0 and ρ∈?. The study of such processes is a time-dependent variation on a classical problem of Lamperti; moreover, they arise naturally in the context of the distance of simple random walk on ? ^d from its centre of mass, for which we also give an apparently new result. We give a recurrence classification and asymptotic theory for processes Z _n satisfying (0.1), which enables us to deduce the complete recurrence classification (for any β≥0) of X _n ?G _n for our self-interacting walk.

     

Evolution of a Modified Binomial Random Graph by Agglomeration

In the classical Erd?s–Rényi random graph G(n, p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n, p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erd?s-Rényi random graph G(n, p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n, p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n, p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real-world applications.     

Alzheimer random walk

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution \(b{\left( {\tfrac{a}{b}} \right)^{\tfrac{{a + 1}}{b}}}{\left[ {\Gamma \left( {\tfrac{{a + 1}}{b}} \right)} \right]^{ - 1}}{x^a}\exp \left( { - \tfrac{a}{b}{x^b}} \right)\) where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p ^{? α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) ? ν(0) ] / [ ν(1) ? ν(0) ] = p ^β with β = 0.388 when p ? 0.1 and β = 0.0822 when p ? 0.1.     

Two-Dimensional Anisotropic Random Walks: Fixed Versus Random Column Configurations for Transport Phenomena

We consider random walks on the square lattice of the plane along the lines of Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) and den Hollander (J Stat Phys 75:891–918, 1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions á  la Heyde (J Stat Phys 27:721–730, 1982, Stochastic processes, Springer, New York, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (J Stat Phys 75:891–918, 1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context.