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Malwina Luczak

Publications and source records attributed to Malwina Luczak.

At least 19 recordsLinked to original sources
arXiv2026

Convergence to equilibrium for density dependent Markov jump processes

We investigate the convergence to (quasi--)equilibrium of a density dependent Markov chain in~${\mathbb Z}^d$, whose drift satisfies a system of ordinary differential equations having an attractive fixed point. For a sequence of such processes~${\mathbb X}^N$, indexed by a size parameter~$N$, the time taken until the distribution of~${\mathbb X}^N$, started in some given state, approaches its (quasi--)equilibrium distribution~$π^N$ typically increases with~$N$. To first order, it corresponds to the time~$t_N$ at which the solution to the drift equations reaches a distance of~$\sqrt N$ from their fixed point. However, the length of the time interval over which the total variation distance between ${\mathcal L} ({\mathbb X}^N(t))$ and its (quasi--)equilibrium distribution~$π^N$ changes from being close to~$1$ to being close to zero is asymptotically of smaller order than~$t_N$. In this sense, the chains exhibit `cut--off', and we are able to prove that the cut-off window is of (optimal) constant size.

math.PR↗
arXiv2025

Cutoff for the logistic SIS epidemic model with self-infection

We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ``self-infection'' is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time, and show the window size is constant (optimal) order. While this result is interesting in its own right, an additional contribution of our work is that the proof illustrates a recently formalised methodology of Barbour, Brightwell and Luczak, which can be used to show cutoff via a combination of concentration of measure inequalities for the trajectory of the chain, and coupling techniques.

math.PR↗
arXiv2022

Long-term concentration of measure and cut-off

We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains. We apply our discrete-time inequality to the well-studied Bernoulli-Laplace model of diffusion, and give a probabilistic proof of cut-off, recovering and improving the bounds of Diaconis and Shahshahani. We also extend the notion of cut-off to chains with an infinite state space, and illustrate this in a second example, of a two-host model of disease in continuous time. We give a third example, giving concentration results for the supermarket model, illustrating the full generality and power of our results.

math.PR↗
arXiv2018

Extinction times in the subcritical stochastic SIS logistic epidemic

Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size $N$. We study the behaviour of the process as the population size $N$ tends to infinity. Our results cover the entire subcritical regime, including the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to 0 as $N \to \infty$ but more slowly than $N^{-1/2}$. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory.

math.PR↗
arXiv2018

Extinction time for the weaker of two competing SIS epidemics

We consider a simple stochastic model for the spread of a disease caused by two virus strains in a closed homogeneously mixing population of size N. The spread of each strain in the absence of the other one is described by the stochastic logistic SIS epidemic process, and we assume that there is perfect cross-immunity between the two strains, that is, individuals infected by one are temporarily immune to re-infections and infections by the other. For the case where one strain has a strictly larger basic reproductive ratio than the other, and the stronger strain on its own is supercritical (that is, its basic reproductive ratio is larger than 1), we derive precise asymptotic results for the distribution of the time when the weaker strain disappears from the population, that is, its extinction time. We further extend our results to certain parameter values where the difference between the two reproductive ratios may tend to 0 as $N \to \infty$. In proving our results, we illustrate a new approach to a fluid limit approximation for a sequence of Markov chains in the vicinity of a stable fixed point of the limit.

math.PR↗
arXiv2016

Component structure of the configuration model: barely supercritical case

We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $ν_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $ν_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=ν_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n ρ_n {\mathbb E} D_n (1+o(1))$, where $ρ_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \emph{critical} regime, where $\varepsilon_n = O(n^{-1/3} ({\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\cite{HatamiMolloy}.

math.PR↗
arXiv2015

Near-critical SIR epidemic on a random graph with given degrees

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered (SIR) epidemic on a random (multi)graph on $n$ vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is $1 + ω(n) n^{-1/3}$, with $ω(n)$ tending to infinity slowly as the population size, $n$, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as $n \to \infty$. As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the `sparse' regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

math.PR↗
arXiv2015

The greedy independent set in a random graph with given degrees

We analyse the size of an independent set in a random graph on $n$ vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as $n \to \infty$, expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges to the jamming constant as $n \to \infty$. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model.

math.PR↗
arXiv2015

The mathematics of causal sets

The causal set approach to quantum gravity is based on the hypothesis that the underlying structure of spacetime is that of a random partial order. We survey some of the interesting mathematics that has arisen in connection with the causal set hypothesis, and describe how the mathematical theory can be translated to the application area. We highlight a number of open problems of interest to those working in causal set theory.

math.CO↗
arXiv2014

Law of large numbers for the SIR epidemic on a random graph with given degrees

We study the susceptible-infective-recovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak occurs with probability bounded away from zero. Our main result is that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. We also consider more general initial conditions for the epidemic, and derive criteria for a simple vaccination strategy to be successful. In contrast to earlier results for this model, our approach only requires basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex. En route, we prove analogous results for the epidemic on the configuration model multigraph under much weaker conditions. Essentially, our main result requires only that the initial values for our processes converge, i.e. it is the best possible.

math.PR↗
arXiv2014

Individual and patch behaviour in structured metapopulation models

Density dependent Markov population processes with countably many types can often be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, provided that the total population size is large. They also exhibit diffusive stochastic fluctuations on a smaller scale about this deterministic path. Here, it is shown that the individuals in such processes experience an almost deterministic environment. Small groups of individuals behave almost independently of one another, evolving as Markov jump processes, whose transition rates are prescribed functions of time. In the context of metapopulation models, we show that `individuals' can represent either patches or the individuals that migrate among the patches; in host--parasite systems, they can represent both hosts and parasites.

math.PR↗
arXiv2013

A quantitative differential equation approximation for a routing model

We consider a Markovian load balancing model on a fully-connected network, where calls have Poisson arrivals and exponential durations. The endpoints of each call are uniform over all the links of the network. Each call is routed either along the link connecting its endpoints, or, if the direct route is unavailable, along a two-link path between them, via an intermediate node. We use an explicit and simple coupling to show a strong concentration of measure property, and deduce that the evolution of the process may be approximated by a differential equation. The technique is likely to be useful in other settings.

math.PR↗
arXiv2013

A fixed-point approximation for a routing model in equilibrium

We use a method of Luczak (arXiv:1212.3231) to investigate the equilibrium distribution of a dynamic routing model on a network. In this model, there are $n$ nodes, each pair joined by a link of capacity $C$. For each pair of nodes, calls arrive for this pair of endpoints as a Poisson process with rate $λ$. A call for endpoints $\{u,v\}$ is routed directly onto the link between the two nodes if there is spare capacity; otherwise $d$ two-link paths between $u$ and $v$ are considered, and the call is routed along a path with lowest maximum load, if possible. The duration of each call is an exponential random variable with unit mean. In the case $d=1$, it was suggested by Gibbens, Hunt and Kelly in 1990 that the equilibrium of this process is related to the fixed points of a certain equation. We show that this is indeed the case, for every $d \ge 1$, provided the arrival rate $λ$ is either sufficiently small or sufficiently large. In either regime, we show that the equation has a unique fixed point, and that, in equilibrium, for each $j$, the proportion of links at each node with load $j$ is strongly concentrated around the $j$th coordinate of the fixed point.

math.PR↗
arXiv2012

Order-invariant measures on fixed causal sets

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers -- we call such a linear extension a {\em natural extension}. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of {\em order-invariance}: if we condition on the set of the bottom $k$ elements of the natural extension, each possible ordering among these $k$ elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

math.CO↗
arXiv2012

The supermarket model with arrival rate tending to one

In the supermarket model, there are $n$ queues, each with a single server. Customers arrive in a Poisson process with arrival rate $λn$, where $λ= λ(n) \in (0,1)$. Upon arrival, a customer selects $d=d(n)$ servers uniformly at random, and joins the queue of a least-loaded server amongst those chosen. Service times are independent exponentially distributed random variables with mean~1. In this paper, we analyse the behaviour of the supermarket model in a regime where $λ(n)$ tends to~1, and $d(n)$ tends to infinity, as $n \to \infty$. For suitable triples $(n,d,λ)$, we identify a subset ${\cal N}$ of the state space where the process remains for a long time in equilibrium. We further show that the process is rapidly mixing when started in ${\cal N}$, and give bounds on the speed of mixing for more general initial conditions.

math.PR↗
arXiv2011

Order-invariant measures on causal sets

A causal set is a partially ordered set on a countably infinite ground-set such that each element is above finitely many others. A natural extension of a causal set is an enumeration of its elements which respects the order. We bring together two different classes of random processes. In one class, we are given a fixed causal set, and we consider random natural extensions of this causal set: we think of the random enumeration as being generated one point at a time. In the other class of processes, we generate a random causal set, working from the bottom up, adding one new maximal element at each stage. Processes of both types can exhibit a property called order-invariance: if we stop the process after some fixed number of steps, then, conditioned on the structure of the causal set, every possible order of generation of its elements is equally likely. We develop a framework for the study of order-invariance which includes both types of example: order-invariance is then a property of probability measures on a certain space. Our main result is a description of the extremal order-invariant measures.

math.PR↗
arXiv2007

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n tend to infinity. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np=1+omega(n)n^{-1/3}, where omega(n) tends to infinity arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs.

math.CO↗