Metastability and the Ising model.
Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form on or subsets of , where is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of can be related, up to multiplicative errors that tend to one as , to the capacities of suitably constructed sets. We show that these capacities...
Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues
We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form on or subsets of , where is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum...
Metastability of the three dimensional Ising model on a torus at very low temperatures.
Metric rigidity of crystallographic groups.
Microscopic concavity and fluctuation bounds in a class of deposition processes
We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude t1/3. This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes....
Microscopic irreversibility.
Microscopic structure of a decreasing shock for the asymmetric -step exclusion process.
Microtheoretical and numerical calculation of phonon spectra in superlattices.
Minimizers of the Lawrence–Doniach energy in the small-coupling limit : finite width samples in a parallel field
Miura opers and critical points of master functions
Critical points of a master function associated to a simple Lie algebra come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms...
Mixed methods for the approximation of liquid crystal flows
The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.
Mixed Methods for the Approximation of Liquid Crystal Flows
The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen–Leslie equations is considered. Convergence results for this system rely crucially upon energy estimates which involve H2(Ω) norms of the director field. We show how a mixed method may be used to eliminate the need for Hermite finite elements and establish convergence of the method.
Mixing time for the Ising model : a uniform lower bound for all graphs
Consider Glauber dynamics for the Ising model on a graph of n vertices. Hayes and Sinclair showed that the mixing time for this dynamics is at least nlog n/f(Δ), where Δ is the maximum degree and f(Δ) = Θ(Δlog2Δ). Their result applies to more general spin systems, and in that generality, they showed that some dependence on Δ is necessary. In this paper, we focus on the ferromagnetic Ising model and prove that the mixing time of Glauber dynamics on any n-vertex graph is at least (1/4 + o(1))nlog n....
Mobility and Einstein relation for a tagged particle in asymmetric mean zero random walk with simple exclusion
Modèles cinétiques à répartition discrète des vitesses et à invariance galiléenne
Modeling a quantum Hall system via elliptic equations.
Modeling Morphogenesis in silico and in vitro: Towards Quantitative, Predictive, Cell-based Modeling
Cell-based, mathematical models help make sense of morphogenesis—i.e. cells organizing into shape and pattern—by capturing cell behavior in simple, purely descriptive models. Cell-based models then predict the tissue-level patterns the cells produce collectively. The first step in a cell-based modeling approach is to isolate sub-processes, e.g. the patterning capabilities of one or a few cell types in cell cultures. Cell-based models can then identify the mechanisms responsible for patterning in...
Modeling snow crystal growth. I: Rigorous results for Packard's digital snowflakes.
Modelli di Spin deterministici con comportamento vetroso