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Motivated by the psychological literature on the "peak-end rule" for remembered experience, we perform an analysis within a random walk framework of a discrete choice model where agents' future choices depend on the peak memory of their…
We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of…
Macroevolutionary dynamics often display sudden, explosive surges, where systems remain relatively stable for extended periods before experiencing dramatic acceleration that frequently exceeds traditional exponential growth. This pattern is…
Diffusion processes are central to human interactions. One common prediction of the current modeling frameworks is that initial spreading dynamics follow exponential growth. Here, we find that, ranging from mobile handsets to automobiles,…
A critical issue in the evolution of software models is change propagation: given a primary change that is made to a model in order to meet a new or changed requirement, what additional secondary changes are needed to maintain consistency…
The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a \emph{time-dependent random medium}, describing the…
We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle…
The Bianconi-Barabasi model of a growing network is revisited. This model, defined by a preferential attachment rule involving both the degrees of the nodes and their intrinsic fitnesses, has the fundamental property to undergo a phase…
Consider Dyson's Hermitian Brownian motion model after a finite time S, where the process is started at N equidistant points on the real line. These N points after time S form a determinantal process and has a limit as N tends to infinity.…
Is evolution always gradual or can it make leaps? We examine a mathematical model of an evolutionary process on a fitness landscape and obtain analytic solutions for the probability of multi-mutation leaps, that is, several mutations…
The modelling of evolutionary game dynamics in finite populations requires microscopic processes that determine how strategies spread. The exact details of these processes are often chosen without much further consideration. Different types…
In a laboratory experiment, round by round, individual interactions should lead to the social evolutionary rotation in population strategy state space. Successive switching the incentive parameter should lead to successive change of the…
The environment in which a population evolves can have a crucial impact on selection. We study evolutionary dynamics in finite populations of fixed size in a changing environment. The population dynamics are driven by birth and death…
It has long been noticed that high dimension data exhibits strange patterns. This has been variously interpreted as either a "blessing" or a "curse", causing uncomfortable inconsistencies in the literature. We propose that these patterns…
Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course…
We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle…
We consider an interacting particle system on the one dimensional lattice $\bf Z$ modeling combustion. The process depends on two integer parameters $2\le a<M<\infty$. Particles move independently as continuous time simple symmetric random…
We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. L\'{e}vy processes endowed with rank-dependent drift and diffusion coefficients. In…
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…
The extreme value statistics of active matter offer significant insight into their unique properties. A phase transition has recently been reported in a model of branching run-and-tumble particles, describing the spatial spreading of an…