We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators form -\epsilon \Delta +\nabla F(\cdot)\nabla on \mathbb R^d or subsets , where F is smooth function finitely many local minima. In analogy previous work discrete Markov chains, we show that metastable exit times from attractive domains minima can be related, up multiplicative errors tend one as \epsilon\downarrow 0 capacities suitably constructed sets. this computed,...

We continue the analysis of problem metastability for reversible diffusion processes, initiated in [BEGK3], with a precise low-lying spectrum generator. Recall that we are considering processes generators form -\epsilon \Delta +\nabla F(\cdot)\nabla on \R^d or subsets , where F is smooth function finitely many local minima. Here consider only generic situation depths all minima different. show general exponentially small part given, up to multiplicative errors tending one, by eigenvalues...

Abstract Branching Brownian motion describes a system of particles that diffuse in space and split into offspring according to certain random mechanism. By virtue the groundbreaking work by M. Bramson on convergence solutions Fisher‐KPP equation traveling waves, law rightmost particle limit large times is rather well understood. In this work, we address full statistics extremal (first‐, second‐, third‐largest, etc.). particular, prove t ‐limit, such descend with overwhelming probability from...

As a first step toward characterization of the limiting extremal process branching Brownian motion, we proved in recent work [Comm. Pure Appl. Math. 64 (2011) 1647–1676] that, limit large time $t$, particles descend with overwhelming probability from ancestors having split either within distance order $1$ $0$, or 1 $t$. The result suggests that motion is randomly shifted cluster point process. Here put part this picture on rigorous ground: prove obtained by retaining only those which are...

Abstract We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range biological applications. Our primary motivation is modelling immunotherapy malignant tumours. In this context different actors, T-cells, cytokines or cancer cells, are modelled as single particles (individuals) system. The main expansions distinguishing cells by phenotype and genotype, including environment-dependent phenotypic plasticity that does not affect...

We consider the random fluctuations of free energy in $p$-spin version Sherrington–Kirkpatrick (SK) model high-temperature regime. Using martingale approach Comets and Neveu as used standard SK combined with truncation techniques inspired by a recent paper Talagrand on version, we prove that corrections to are scale $N^{-(p-2)/2}$ only and, after proper rescaling, converge Gaussian variable. This is shown hold for all values inverse temperature, $\beta$, smaller than critical $\beta_p$. also...

The authors study a one-dimensional tight binding model with random potential taking two values +or- nu , the restriction that on pairs of neighbouring sites potentials take same value. This dimer model, proposed by Dunlap, Phillips and Wu (1990), has vanishing Lyapunov exponent for energy E=+or- . They compute density states perturbatively in vicinity this using invariant measure formalism.

We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang Zeitouni, case of piecewise constant speeds; in fact simplicity we concentrate on when is $\sigma_1$ $s\leq bt$ $\sigma_2$ $bt\leq s\leq t$. In $\sigma_1>\sigma_2$, concatenation two BBM processes, as expected. $\sigma_1<\sigma_2$, a new family cluster point processes arises, that are similar, but distinctively different from process. Our proofs follow strategy Arguin,...

This is the second of a series three papers in which we present rigorous analysis Derrida's Generalized Random Energy Models (GREM). Here study general case models with "continuum hierarchies". We prove convergence free energy and give explicit formulas for two-replica distribution function thermodynamical limit. Then introduce empirical distance to describe effectively Gibbs measures. show that its limit uniquely determined via Ghirlanda–Guerra identities up mean replica function. Finally,...

The random energy model (REM) has become a key reference for glassy systems. In particular, it is expected to provide prime example of system whose dynamics shows aging, universal phenomenon characterizing the complex analysis its activated based on so-called trap models, introduced by Bouchaud, that are also used mimic more disordered this Letter we report first results justify rigorously predictions in REM.