arXiv subjects

Frank den Hollander

Publications and source records attributed to Frank den Hollander.

At least 37 records · Page 2Linked to original sources

arXiv2025

The voter model on random regular graphs with random rewiring

We consider the voter model with binary opinions on a random regular graph with $n$ vertices of degree $d \geq 3$, subject to a rewiring dynamics in which pairs of edges are rewired, i.e., broken into four half-edges and subsequently reconnected at random. A parameter $ν\in (0,\infty)$ regulates the frequency at which the rewirings take place, in such a way that any given edge is rewired exponentially at a rate $ν$ in the limit as $n\to\infty$. We show that, under the joint law of the random rewiring dynamics and the random opinion dynamics, the fraction of vertices with either one of the two opinions converges on time scale $n$ to the Fisher-Wright diffusion with an explicit diffusion constant $\vartheta_{d,ν}$ in the limit as $n\to\infty$. In particular, we identify $\vartheta_{d,ν}$ in terms of a continued-fraction expansion and analyse its dependence on $d$ and $ν$. A key role in our analysis is played by the set of discordant edges, which constitutes the boundary between the sets of vertices carrying the two opinions.

math.PR↗

arXiv2025

Large deviation principle for the norm of the Laplacian matrix of inhomogeneous Erdős-Rényi random graphs

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r_N(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r_N\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the Laplacian matrix of $G_N$. We show that if $\lim_{N\to\infty} \|r_N-r\|_\infty = 0$ for some limiting graphon $r\colon\,[0,1]^2 \to (0,1)$, then $λ_N/N$ satisfies a downward LDP with rate $\binom{N}{2}$ and an upward LDP with rate $N$. We identify the associated rate functions $ψ_r$ and $\widehatψ_r$, and derive their basic properties.

math.PR↗

arXiv2024

A Unifying Theory of Aging and Mortality

In this paper, we advance the network theory of aging and mortality by developing a causal mathematical model for the mortality rate. First, we show that in large networks, where health deficits accumulate at nodes representing health indicators, the modeling of network evolution with Poisson processes is universal and can be derived from fundamental principles. Second, with the help of two simplifying approximations, which we refer to as mean-field assumption and homogeneity assumption, we provide an analytical derivation of Gompertz law under generic and biologically relevant conditions. We identify the parameters in Gompertz law as a function of the parameters driving the evolution of the network, and illustrate our computations with simulations and analytic approximations.

q-bio.QM↗

arXiv2024

The Moran model with random resampling rates

In this paper we consider the two-type Moran model with $N$ individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution ${\mathbb P}$ on ${\mathbb R}_+$, and a type, either $1$ or $0$. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let $Y^N(t)$ denote the empirical distribution of the resampling rates of the individuals with type $1$ at time $Nt$. We show that if ${\mathbb P}$ has countable support and satisfies certain tail and moment conditions, then in the limit as $N\to\infty$ the process $(Y^N(t))_{t \geq 0}$ converges in law to the process $(S(t)\,¶)_{t \geq 0}$, in the so-called Meyer-Zheng topology, where $(S(t))_{t \geq 0}$ is the Fisher-Wright diffusion with diffusion constant $D$ given by $1/D = \int_{{\mathbb R}_+} (1/r)\,{\mathbb P}(\mathrm{d} r)$.

math.PR↗

arXiv2024

Opinion dynamics on dense dynamic random graphs

We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models.

math.PR↗

arXiv2024

Metastability of Glauber dynamics with inhomogeneous coupling disorder

We introduce a general class of mean-field-like spin systems with random couplings that comprises both the Ising model on inhomogeneous dense random graphs and the randomly diluted Hopfield model. We are interested in quantitative estimates of metastability in large volumes at fixed temperatures when these systems evolve according to a Glauber dynamics, i.e.\ where spins flip with Metropolis transition probabilities at inverse temperature $β$. We identify conditions ensuring that with high probability the system behaves like the corresponding system where the random couplings are replaced by their averages. More precisely, we prove that the metastability of the former system is implied with high probability by the metastability of the latter. Moreover, we consider relevant metastable hitting times of the two systems and find the asymptotic tail behaviour and the moments of their ratio. This work provides an extension of the results known for the Ising model on the Erdős--R{é}nyi random graph. The proofs use the potential-theoretic approach to metastability in combination with concentration inequalities.

math.PR↗

arXiv2024

Droplet dynamics in a two-dimensional rarefied gas under Kawasaki dynamics

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature $β>0$ in a large finite box $Λ_β\subset\mathbb Z^2$ whose volume depends on $β$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $Δ>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $Δ\in(U,2U)$ and in the limit as $β\to\infty$, how and when the system nucleates. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, i.e., $|Λ_β|=\mathrm e^{Θβ}$ with $Δ<Θ<2Δ-U$. In the third paper we consider the setting where the volume is very large, namely, $|Λ_β|=\mathrm e^{Θβ}$ with $Δ<Θ<Γ-(2Δ-U)$, where $Γ$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach: the tube of typical trajectories leading to nucleation is described via a series of events on which the evolution of the gas consists of droplets wandering around on multiple space-time scales.

math.PR↗

arXiv2024

Discordant edges for the voter model on regular random graphs

We consider the two-opinion voter model on a regular random graph with n vertices and degree $d \geq 3$. It is known that consensus is reached on time scale n and that on this time scale the volume of the set of vertices with one opinion evolves as a Fisher-Wright diffusion. We are interested in the evolution of the number of discordant edges (i.e., edges linking vertices with different opinions), which can be thought as the perimeter of the set of vertices with one opinion, and is the key observable capturing how consensus is reached. We show that if initially the two opinions are drawn independently from a Bernoulli distribution with parameter $u \in (0, 1)$, then on time scale 1 the fraction of discordant edges decreases and stabilises to a value that depends on d and u, and is related to the meeting time of two random walks on an infinite tree of degree d starting from two neighbouring vertices. Moreover, we show that on time scale n the fraction of discordant edges moves away from the constant plateau and converges to zero in an exponential fashion. Our proofs exploit the classical dual system of coalescing random walks and use ideas from Cooper et al. (2010) built on the so-called First Visit Time Lemma. We further introduce a novel technique to derive concentration properties from weak-dependence of coalescing random walks on moderate time scales.

math.PR↗

arXiv2024

Mixing of fast random walks on dynamic random permutations

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one allows for coagulation of permutation cycles only, the other allows for both coagulation and fragmentation. We show that for both types, after scaling time by the length of the permutation and letting this length tend to infinity, the total variation distance between the current distribution and the uniform distribution converges to a limit process that drops down in a single jump. This jump is similar to a one-sided cut-off, occurs after a random time whose law we identify, and goes from the value 1 to a value that is a strictly decreasing and deterministic function of the time of the jump, related to the size of the largest component in Erdős-Rényi random graphs. After the jump, the total variation distance follows this function down to 0.

math.PR↗

arXiv2023

Largest Eigenvalue of the Configuration Model and Breaking of Ensemble Equivalence

We analyse the largest eigenvalue of the adjacency matrix of the configuration model with large degrees, where the latter are treated as hard constraints. In particular, we compute the expectation of the largest eigenvalue for degrees that diverge as the number of vertices $n$ tends to infinity, uniformly on a scale between $1$ and $\sqrt{n}$, and show that a weak law of large numbers holds. We compare with what was derived in our earlier paper "Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs" for the Chung-Lu model, which in the regime considered represents the corresponding configuration model with soft constraints, and show that the expectation is shifted down by $1$ asymptotically. This shift is a signature of breaking of ensemble equivalence between the hard and soft (also known as micro-canonical and canonical) versions of the configuration model. The latter result generalizes a previous finding in "A spectral signature of breaking of ensemble equivalence for constrained random graphs" obtained in the case when all degrees are equal.

math.PR↗

arXiv2023

Quenched large deviations in renewal theory

In this paper we introduce and study renewal-reward processes in random environments where each renewal involves a reward taking values in a Banach space. We derive quenched large deviation principles and identify the associated rate functions in terms of variational formulas involving correctors. We illustrate the theory with three examples: compound Poisson processes in random environments, pinning of polymers at interfaces with disorder, and returns of Markov chains in dynamic random environments.

math.PR↗

arXiv2023

Wireless random-access networks with bipartite interference graphs

We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

math.PR↗

arXiv2023

The grapheme-valued Wright-Fisher diffusion with mutation

In [Athreya, den Hollander, Röllin; 2021, arXiv:1908.06241] models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterised as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set, an adjacency matrix and a sampling measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Poisson-Dirichlet distribution. In a companion paper [Greven, den Hollander, Klimovsky, Winter; 2023], we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.

math.PR↗

arXiv2023

The annealed parabolic Anderson model on a regular tree

We study the total mass of the solution to the parabolic Anderson model on a regular tree with an i.i.d. random potential whose marginal distribution is double-exponential. In earlier work we identified two terms in the asymptotic expansion for large time of the total mass under the quenched law, i.e., conditional on the realisation of the random potential. In the present paper we do the same for the annealed law, i.e., averaged over the random potential. It turns out that the annealed expansion differs from the quenched expansion. The derivation of the annealed expansion is based on a new approach to control the local times of the random walk appearing in the Feynman-Kac formula for the total mass. In particular, we condition on the backbone to infinity of the random walk, truncate and periodise the infinite tree relative to the backbone to obtain a random walk on a finite subtree with a specific boundary condition, employ the large deviation principle for the empirical distribution of Markov renewal processes on finite graphs, and afterwards let the truncation level tend to infinity to obtain an asymptotically sharp asymptotic expansion.

math.PR↗

arXiv2022

Breaking of ensemble equivalence for dense random graphs under a single constraint

Two ensembles are often used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. The latter means that it matters for the scaling properties of the graph whether the constraint is met for every single realisation of the graph or only holds as an ensemble average. In the literature, it was found that BEE is the rule rather than the exception for two classes: sparse random graphs when the number of constraints is of the order of the number of vertices and dense random graphs when there are two or more constraints that are frustrated. In the present paper we establish BEE for a third class: dense random graphs with a single constraint, namely, on the density of a given finite simple graph. We show that BEE occurs only in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as spectral signature of BEE. Proofs are based on an analysis of the variational formula on the space of graphons for the limiting specific relative entropy derived by Den Hollander et al. (2018), in combination with an identification of the minimising graphons and replica symmetry arguments. We show that in the replica symmetric region of the BEE-phase, as the size of the graph tends to infinity, the microcanonical ensemble behaves like an Erdős-Rényi random graph, while the canonical ensemble behaves like a mixture of two Erdős-Rényi random graphs.

math.PR↗

arXiv2022

Quenched large deviation principle for words in a letter sequence

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 \emph{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 \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an \emph{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. The previous version (arXiv:0807.2611v2) appeared in Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403--456. Meanwhile, it has turned out that the original proof of the representation of the rate function is flawed when the mean word length is infinite. We add an erratum in which we fix the flaw in the proof. Along the way we derive new representations of the rate function that are interesting in their own right. A key ingredient in the proof is the observation that if the rate function in the annealed large deviation principle is finite at a stationary word process, then the letters in the tail of the long words in this process are typical.

math.PR↗

arXiv2022

Spatial populations with seed-bank: finite-systems scheme

We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type one of two types, live in colonies, and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group endowed with the discrete topology. In earlier work we showed that the system has a one-parameter family of equilibria controlled by the relative density of the two types. Moreover, these equilibria exhibit a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. If the wake-up time has finite mean, then there is a single universality class for the scaling limit. On the other hand, if the wake-up time has infinite mean, then there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space.

math.PR↗

arXiv2022

Graphon-valued processes with vertex-level fluctuations

We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself.

math.PR↗