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- ItemHydrodynamic limit for the A + B → Ø model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Bovier, Anton; Černý, JiriWe study a two-species interacting particle model on a subset of $Z$ with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an ``order book'' on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation.
- ItemHomogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bovier, Anton; Hollander, Frank den; Spitoni, CristianIn this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $b$ denote the inverse temperature and let $L_b subset Z^2$ be a square box with periodic boundary conditions such that $lim_btoinfty L_b =infty$. We run the dynamics on $L_b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles) are small. For large $b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single emphcritical droplet somewhere in $L_b$. Using potential-theoretic methods, we compute the emphaverage nucleation time (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $btoinfty$. It turns out that this time grows as $Ke^Gammab/ L_b $ for Glauber dynamics and $Kb e^Gammab/ L_b $ for Kawasaki dynamics, where $Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $ L_b $). The fact that the average nucleation time is inversely proportional to $ L_b $ is referred to as emphhomogeneous nucleation, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $L_b$.
- ItemUniversality of the REM for dynamics of mean-field spin glasses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Ben Arous, Gérard; Bovier, Anton; Černý, JiřiWe consider a version of a Glauber dynamics for a $p$-spin Sherrington--Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the $N$-dimensional hypercube. We show that, for any $p geq 3$ and any inverse temperature $beta>0$, there exist constants $g_0>0$, such that for all exponential time scales, $exp(gamma N)$, with $gleq g_0$, the properly rescaled emphclock process (time-change process), converges to an $a$-stable subordinator where $a=g/b^2<1$. Moreover, the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this hbox$alpha$-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system), the dynamics of $p$-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case $p=2$) seems to belong to a different universality class.
- ItemUniversality of REM-like ageing in mean field spin glasses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Ben Arous, G.; Bovier, Anton; Černý, JiříAging has become the paradigm to describe dynamical behavior of glassy systems, and in particular spin glasses. Trap models have been introduced as simple caricatures of effective dynamics of such systems. In this Letter we show that in a wide class of mean field models and on a wide range of time scales, aging occurs precisely as predicted by the REM-like trap model of Bouchaud and Dean. This is the first rigorous result about aging in mean field models except for the REM and the spherical model
- ItemMetastability : a potential theoretic approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2005) Bovier, AntonMetastability is an ubiquitous phenomenon of the dynamical behaviour of complex systems. In this talk, I describe recent attempts towards a model-independent approach to metastability in the context of reversible Markov processes. I will present an outline of a general theory, based on careful use of potential theoretic ideas and indicate a number of concrete examples where this theory was used very successfully. I will also indicate some challenges for future work.
- ItemSharp asymptotics for metastability in the random field Curie-Weiss model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bianchi, Alessandra; Bovier, Anton; Ioffe, DmitryIn this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.