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Peter Morters

Publications and source records attributed to Peter Morters.

15 recordsLinked to original sources
arXiv2017

Near critical preferential attachment networks have small giant components

Preferential attachment networks with power law exponent $τ>3$ are known to exhibit a phase transition. There is a value $ρ_{\rm c}>0$ such that, for small edge densities $ρ\leq ρ_c$ every component of the graph comprises an asymptotically vanishing proportion of vertices, while for large edge densities $ρ>ρ_c$ there is a unique giant component comprising an asymptotically positive proportion of vertices. In this paper we study the decay in the size of the giant component as the critical edge density is approached from above. We show that the size decays very rapidly, like $\exp(-c/ \sqrt{ρ-ρ_c})$ for an explicit constant $c>0$ depending on the model implementation. This result is in contrast to the behaviour of the class of rank-one models of scale-free networks, including the configuration model, where the decay is polynomial. Our proofs rely on the local neighbourhood approximations of [Dereich, Morters, 2013] and recent progress in the theory of branching random walks [Gantert, Hu, Shi, 2011].

math.PR↗
arXiv2017

Non-extensive condensation in reinforced branching processes

We study a class of branching processes in which a population consists of immortal individuals equipped with a fitness value. Individuals produce offspring with a rate given by their fitness, and offspring may either belong to the same family, sharing the fitness of their parent, or be founders of new families, with a fitness sampled from a fitness distribution. Examples that can be embedded in this class are stochastic house-of-cards models, urn models with reinforcement, and the preferential attachment tree of Bianconi and Barabasi. Our focus is on the case when the fitness distribution has bounded support and regularly varying tail at the essential supremum. In this case there exists a condensation phase, in which asymptotically a proportion of mass in the empirical fitness distribution of the overall population condenses in the maximal fitness value. Our main results describe the asymptotic behaviour of the size and fitness of the largest family at a given time. In particular, we show that as time goes to infinity the size of the largest family is always negligible compared to the overall population size. This implies that condensation, when it arises, is non-extensive and emerges as a collective effort of several families none of which can create a condensate on its own. Our result disproves claims made in the physics literature in the context of preferential attachment trees.

math.PR↗
arXiv2016

Optimal embeddings by unbiased shifts of Brownian motion

An unbiased shift of the two-sided Brownian motion $(B_t \colon t\in{\mathbb R})$ is a random time $T$ such that $(B_{T+t} \colon t\in{\mathbb R})$ is still a two-sided Brownian motion. Given a pair $μ, ν$ of orthogonal probability measures, an unbiased shift $T$ solves the embedding problem, if $B_0\simμ$ implies $B_{T}\simν$. A solution to this problem was given by Last et al. (2014), based on earlier work of Bertoin and Le Jan (1992), and Holroyd and Liggett (2001). In this note we show that this solution minimises ${\mathbb E} ψ(T)$ over all nonnegative unbiased solutions $T$, simultaneously for all nonnegative, concave functions $ψ$. Our proof is based on a discrete concavity inequality that may be of independent interest.

math.PR↗
arXiv2015

Skorokhod embeddings for two-sided Markov chains

Let $(X_n \colon n\in\Z)$ be a two-sided recurrent Markov chain with fixed initial state $X_0$ and let $ν$ be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time $T$ such that $(X_{T+n} \colon n\in\Z)$ has the law of the same Markov chain with initial distribution $ν$. In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of $ψ(T)$ in the class of all non-negative solutions, simultaneously for all non-negative concave functions $ψ$.

math.PR↗
arXiv2015

Robustness of scale-free spatial networks

A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the $d$-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent $τ$ of the degree distribution and the rate $δd$ at which the connection probability decreases with the distance of two vertices. We show that the network is robust if $τ<2+1/δ$, but fails to be robust if $τ>3$. In the case of one-dimensional space we also show that the network is not robust if $τ<2+1/(δ-1)$. This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality.

math.PR↗
arXiv2012

A conditioning principle for Galton-Watson trees

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, converges as $\eps\downarrow 0$ in law to the regular $μ$-ary tree, where $μ$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.

math.PR↗
arXiv2012

Galton-Watson trees with vanishing martingale limit

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, agrees up to generation $K$ with a regular $μ$-ary tree, where $μ$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $\log(1/\eps)$. More precisely, we show that if $μ\ge 2$ then with high probability as $\eps \downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $μ$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.

math.PR↗
arXiv2008

Random networks with sublinear preferential attachment: Degree evolutions

We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree.

math.PR↗
arXiv2008

Minimal supporting subtrees for the free energy of polymers on disordered trees

We consider a model of directed polymers on a regular tree with a disorder given by independent, identically distributed weights attached to the vertices. For suitable weight distributions this model undergoes a phase transition with respect to its localization behaviour. We show that, for high temperatures, the free energy is supported by a random tree of positive exponential growth rate, which is strictly smaller than that of the full tree. The growth rate of the minimal supporting subtree is decreasing to zero as the temperature decreases to the critical value. At the critical value and all lower temperatures, a single polymer suffices to support the free energy. Our proofs rely on elegant martingale methods adapted from the theory of branching random walks.

math.PR↗
arXiv2007

Small value probabilities via the branching tree heuristic

In the first part of this paper we give easy and intuitive proofs for the small value probabilities of the martingale limit of a supercritical Galton-Watson process in both the Schröder and the Böttcher case. These results are well-known, but the most cited proofs rely on generating function arguments which are hard to transfer to other settings. In the second part we show that the strategy underlying our proofs can be used in the quite different context of self-intersections of stochastic processes. Solving a problem posed by Wenbo Li, we find the small value probabilities for intersection local times of several Brownian motions, as well as for self-intersection local times of a single Brownian motion.

math.PR↗
arXiv2006

Moderate deviations for random walk in random scenery

We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramer's condition. We prove moderate deviation principles in dimensions two and larger, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case of dimension four and larger we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In dimension three and larger, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in dimension two we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

math.PR↗
arXiv2006

Complete localisation in the parabolic Anderson model with Pareto-distributed potential

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=Δu(t,z)+ξ(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(ξ(z) \colon z\in {\mathbb Z}^d)$. We consider independent and identically distributed potential variables, such that Prob$(ξ(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large time $t$, the solution is completely localised in a single point with high probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$ with values in $\Z^d$ such that $\lim_{t \uparrow\infty} u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.

math.PR↗
arXiv2005

The multifractal spectrum of Brownian intersection local times

Let \ell be the projected intersection local time of two independent Brownian paths in R^d for d=2,3. We determine the lower tail of the random variable \ell(U), where U is the unit ball. The answer is given in terms of intersection exponents, which are explicitly known in the case of planar Brownian motion. We use this result to obtain the multifractal spectrum, or spectrum of thin points, for the intersection local times.

math.PR↗
arXiv2003

A large-deviation theorem for tree-indexed Markov chains

Given a finite typed rooted tree $T$ with $n$ vertices, the {\em empirical subtree measure} is the uniform measure on the $n$ typed subtrees of $T$ formed by taking all descendants of a single vertex. We prove a large deviation principle in $n$, with explicit rate function, for the empirical subtree measures of multitype Galton-Watson trees conditioned to have exactly $n$ vertices. In the process, we extend the notions of shift-invariance and specific relative entropy--as typically understood for Markov fields on deterministic graphs such as $\mathbb Z^d$--to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with $n$ vertices.

math.PR↗