arXiv subjects

Frank den Hollander

Publications and source records attributed to Frank den Hollander.

At least 73 records · Page 4Linked to original sources

arXiv2019

Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erdős-Rényi Random Graphs

Inhomogeneous Erdős-Rényi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N\,f(\frac{i}{N},\frac{j}{N})$, $1 \leq i \neq j \leq N$, independently of other edges, where $f\colon\,[0,1] \times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $Δ_N$ associated with $\mathbb G_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, it is shown that the empirical spectral distributions of $A_N$ and $Δ_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\,[0,1] \to [0,\infty)$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.

math.PR↗

arXiv2019

Graphon-valued stochastic processes from population genetics

The goal of this paper is to develop a theory of graphon-valued stochastic processes, and to construct and analyse a natural class of such processes arising from population genetics. We consider finite populations where individuals change type according to Wright-Fisher resampling. At any time, each pair of individuals is linked by an edge with a probability that is given by a type-connection matrix, whose entries depend on the current empirical type distribution of the entire population via a fitness function. We show that, in the large-population-size limit and with an appropriate scaling of time, the evolution of the associated adjacency matrix converges to a random process in the space of graphons, driven by the type-connection matrix and the underlying Wright-Fisher diffusion on the multi-type simplex. In the limit as the number of types tends to infinity, the limiting process is driven by the type-connection kernel and the underlying Fleming-Viot diffusion.

math.PR↗

arXiv2019

Random walk in cooling random environment: recurrence versus transience and mixed fluctuations

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. An overarching goal of our paper is to investigate how the behaviour of a random process with a rich correlation structure can be affected by resettings.

math.PR↗

arXiv2019

Properties of additive functionals of Brownian motion with resetting

We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by computing distributions and moments, using a formula that links the generating function with resetting to the generating function without resetting.

math.PR↗

arXiv2018

Covariance structure behind breaking of ensemble equivalence in random graphs

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 conjectured 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.

math-ph↗

arXiv2018

Synchronization of phase oscillators on the hierarchical lattice

Synchronization of neurons forming a network with a hierarchical structure is essential for the brain to be able to function optimally. In this paper we study synchronization of phase oscillators on the most basic example of such a network, namely, the hierarchical lattice. Each site of the lattice carries an oscillator that is subject to noise. Pairs of oscillators interact with each other at a strength that depends on their hierarchical distance, modulated by a sequence of interaction parameters. We look at block averages of the oscillators on successive hierarchical scales, which we think of as block communities. In the limit as the number of oscillators per community tends to infinity, referred to as the hierarchical mean-field limit, we find a separation of time scales, i.e., each block community behaves like a single oscillator evolving on its own time scale. We argue that the evolution of the block communities is given by a renormalized mean-field noisy Kuramoto equation, with a synchronization level that depends on the hierarchical scale of the block community. We find three universality classes for the synchronization levels on successive hierarchical scales, characterized in terms of the sequence of interaction parameters. What makes our model specifically challenging is the non-linearity of the interaction betweenthe oscillators. The main results of our paper therefore come in three parts: (I) a conjecture about the nature of the renormalisation transformation connecting successive hierarchical scales; (II) a truncation approximation that leads to a simplified renormalization transformation; (III) a rigorous analysis of the simplified renormalization transformation. We provide compelling arguments in support of (I) and (II), but a full verification remains an open problem.

cond-mat.dis-nn↗

arXiv2018

Random walks on dynamic configuration models: a trichotomy

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction $α_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to\infty$, when $α_n$ is chosen such that $\lim_{n\to\infty} α_n (\log n)^2 = β\in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1/\sqrt{α_n}$ when $β=\infty$. In the present paper we investigate what happens when $β\in [0,\infty)$. It turns out that the mixing time is of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff when $β\in (0,\infty)$ and a two-sided cutoff when $β=0$. The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by Ben-Hamou and Salez [4], and the regeneration time of first stepping across a rewired edge.

math.PR↗

arXiv2018

Mixing times of random walks on dynamic configuration models

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random graph becomes {\em dynamic}, namely, at each unit of time a fraction $α_n$ of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every $\varepsilon\in(0,1)$ the $\varepsilon$-mixing time of random walk without backtracking grows like $\sqrt{2\log(1/\varepsilon)/\log(1/(1-α_n))}$ as $n \to \infty$, provided that $\lim_{n\to\infty} α_n(\log n)^2=\infty$. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.

math.PR↗

arXiv2018

Metastability of hard-core dynamics on bipartite graphs

We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We obtain a sharp asymptotic estimate for the expected transition time, show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results. In a follow-up paper we will use our results to study the performance of random-access wireless networks.

math.PR↗

arXiv2017

Annealed scaling for a charged polymer in dimensions two and higher

This paper considers an undirected polymer chain on $\mathbb{Z}^d$, $d \geq 2$, with i.i.d.\ random charges attached to its constituent monomers. Each self-intersection of the polymer chain contributes an energy to the interaction Hamiltonian that is equal to the product of the charges of the two monomers that meet. The joint probability distribution for the polymer chain and the charges is given by the Gibbs distribution associated with the interaction Hamiltonian. The object of interest is the \emph{annealed free energy} per monomer in the limit as the length $n$ of the polymer chain tends to infinity. We show that there is a critical curve in the parameter plane spanned by the charge bias and the inverse temperature separating an \emph{extended phase} from a \emph{collapsed phase}. We derive the scaling of the critical curve for small and for large charge bias and the scaling of the annealed free energy for small inverse temperature, which are both anomalous. We show that in a subset of the collapsed phase the polymer chain is \emph{subdiffusive}, namely, on scale $(n/\log n)^{1/(d+2)}$ it moves like a Brownian motion conditioned to stay inside a ball with a deterministic radius and a randomly shifted center. We expect this scaling to hold throughout the collapsed phase. We further expect that in the extended phase the polymer chain scales like a weakly self-avoiding walk. Proofs are based on a detailed analysis for simple random walk of the downward large deviations of the self-intersection local time and the upward large deviations of the range. Part of our scaling results are rough, and we formulate conjectures under which they can be sharpened. The existence of the free energy remains an open problem, which we are able to settle in a subset of the collapsed phase for a subclass of charge distributions.

math-ph↗

arXiv2017

Torsional rigidity for regions with a Brownian boundary

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $Ω\subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in Ω$ of the expected lifetime in $Ω$ of a Brownian motion starting at $x$. In this paper we consider $Ω= T^m \backslash β[0,t]$, the complement of the path $β[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \backslash β[0,t]$ whose inradius is comparable to the largest inradius, while for $m=3$ most of $T^3 \backslash β[0,t]$ contributes. A similar result holds for $m \geq 4$ after the Brownian path is replaced by a shrinking Wiener sausage $W_{r(t)}[0,t]$ of radius $r(t)=o(t^{-1/(m-2)})$, provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of $β[0,t]$ in $R^3$ and $W_1[0,t]$ in $R^m$, $m \geq 4$, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on $T^m$, which has received a lot of attention in the literature in past years.

math.PR↗

arXiv2017

The hierarchical Cannings process in random environment

In an earlier paper, we introduced and studied a system of hierarchically interacting measure-valued random processes which describes a large population of individuals carrying types and living in colonies labelled by the hierarchical group of order $N$. The individuals are subject to migration, resampling on all hierarchical scales simultaneously. Upon resampling, a random positive fraction of the population in a block of colonies inherits the type of a random single individual in that block, which is why we refer to our system as the hierarchical Cannings process. In the present paper, we study a version of the hierarchical Cannings process in random environment, namely, the resampling measures controlling the change of type of individuals in different blocks are chosen randomly with a given mean and are kept fixed in time (= the quenched setting). We give a necessary and sufficient condition under which a multi-type equilibrium is approached (= coexistence) as opposed to a mono-type equilibrium (= clustering). Moreover, in the hierarchical mean-field limit $N \to \infty$, with the help of a renormalization analysis, we obtain a full picture of the space-time scaling behaviour of block averages on all hierarchical scales simultaneously. We show that the $k$-block averages are distributed as the superposition of a Fleming-Viot diffusion with a deterministic volatility constant $d_k$ and a Cannings process with a random jump rate, both depending on $k$. In the random environment, $d_k$ turns out to be smaller than in the homogeneous environment of the same mean. We investigate how $d_k$ scales with $k$. This leads to five universality classes of cluster formation in the mono-type regime. We find that if clustering occurs, then the random environment slows down the growth of the clusters, i.e., enhances the diversity of types.

math.PR↗

arXiv2017

Expansion of percolation critical points for Hamming graphs

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on $H(d,n)$. We show that, for $d \in \mathbb N$ fixed and $n \to \infty$, \begin{equation*} p_c^{(d)}= \dfrac{1}{m} + \dfrac{2d^2-1}{2(d-1)^2}\dfrac{1}{m^2} + O(m^{-3}) + O(m^{-1}V^{-1/3}), \end{equation*} which extends the asymptotics found in \cite{BorChaHofSlaSpe05b} by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In \cite{FedHofHolHul16a} \st{we show that} this formula is a crucial ingredient in the study of critical bond percolation on $H(d,n)$ for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdős-Rényi random graph.

math.PR↗

arXiv2016

Metastability on the hierarchical lattice

We study metastability for Glauber spin-flip dynamics on the $N$-dimensional hierarchical lattice with $n$ hierarchical levels. Each vertex carries an Ising spin that can take the values $-1$ or $+1$. Spins interact with an external magnetic field $h>0$. Pairs of spins interact with each other according to a ferromagnetic pair potential $\vec{J}=\{J_i\}_{i=1}^n$, where $J_i>0$ is the strength of the interaction between spins at hierarchical distance $i$. Spins flip according to a Metropolis dynamics at inverse temperature $β$. In the limit as $β\to\infty$, we analyse the crossover time from the metastable state $\boxminus$ (all spins $-1$) to the stable state $\boxplus$ (all spins $+1$). Under the assumption that $\vec{J}$ is non-increasing, we identify the mean transition time up to a multiplicative factor $1+o_β(1)$. On the scale of its mean, the transition time is exponentially distributed. We also identify the set of configurations representing the gate for the transition. For the special case where $J_i = \tilde{J}/N^i$, $1 \leq i \leq n$, with $\tilde{J}>0$ the relevant formulas simplify considerably. Also the hierarchical mean-field limit $N\to\infty$ can be analysed in detail.

math.PR↗

arXiv2016

Ensemble nonequivalence in random graphs with modular structure

Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In Squartini et al. (2015) it was shown that ensembles of random graphs are non-equivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, block-model or community structure. We give a full classification of ensemble equivalence, proving that breakdown occurs if and only if the number of local constraints (i.e., the number of constrained degrees) is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive a formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.

math.PR↗

arXiv2016

Multi-colony Wright-Fisher with seed-bank

We consider a multi-colony version of the Wright-Fisher model with seed-bank that was recently introduced by Blath et al. Individuals live in colonies and change type via resampling and mutation. Each colony contains a seed-bank that acts as a genetic reservoir. Individuals can enter the seed-bank and become dormant or can exit the seed-bank and become active. In each colony at each generation a fixed fraction of individuals swap state, drawn randomly from the active and the dormant population. While dormant, individuals suspend their resampling. While active, individuals resample from their own colony, but also from different colonies according to a random walk transition kernel representingmigration. Both active and dormant individuals mutate. We derive a formula for the probability that two individuals drawn randomly from two given colonies are identical by descent, i.e., share a common ancestor. This formula, which is formulated in Fourier language, is valid when the colonies form a discrete torus. We consider the special case of a symmetric slow seed-bank, for which in each colony half of the individuals are in the seed-bank and at each generation the fraction of individuals that swap state is small. This leads to a simpler formula, from which we are able to deduce how the probability to be identical by descent depends on the distance between the two colonies and various relevant parameters. Through an analysis of random walk Green functions, we are able to derive explicit scaling expressions when mutation is slower than migration. We also compute the spatial second moment of the probability to be identical by descent for all parameters when the torus becomes large. For the special case of a symmetric slow seed-bank, we again obtain explicit scaling expressions.

math.PR↗

arXiv2016

Phase diagram for a copolymer in a micro-emulsion

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 $\{(α,β) \in\mathbb{R}^2\colon\,α\geq |β|\}$. Depending on the values of these parameters, the copolymer either stays close to the oil-water 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 copolymer can move through the micro-emulsion. The analysis of the phase diagram is based on three hypotheses of percolation-type on the blocks. We show that these three hypotheses are plausible, but do not provide a proof.

math.PR↗

arXiv2016

Metastability for Glauber dynamics on random graphs

In this paper we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the Configuration Model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain \emph{key hypothesis} implying metastable behaviour, the average crossover time follows the classical \emph{Arrhenius law}, with an exponent and a prefactor that are controlled by the \emph{energy} and the \emph{entropy} of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the Configuration Model are expander graphs.

math.PR↗