80 research outputs found
Metastability in the dilute Ising model
Consider Glauber dynamics for the Ising model on the hypercubic lattice with
a positive magnetic field. Starting from the minus configuration, the system
initially settles into a metastable state with negative magnetization. Slowly
the system relaxes to a stable state with positive magnetization. Schonmann and
Shlosman showed that in the two dimensional case the relaxation time is a
simple function of the energy required to create a critical Wulff droplet.
The dilute Ising model is obtained from the regular Ising model by deleting a
fraction of the edges of the underlying graph. In this paper we show that even
an arbitrarily small dilution can dramatically reduce the relaxation time. This
is because of a catalyst effect---rare regions of high dilution speed up the
transition from minus phase to plus phase.Comment: 49 page
Stretched Polymers in Random Environment
We survey recent results and open questions on the ballistic phase of
stretched polymers in both annealed and quenched random environments.Comment: Dedicated to Erwin Bolthausen on the occasion of his 65th birthda
Modeling the scaling properties of human mobility
While the fat tailed jump size and the waiting time distributions
characterizing individual human trajectories strongly suggest the relevance of
the continuous time random walk (CTRW) models of human mobility, no one
seriously believes that human traces are truly random. Given the importance of
human mobility, from epidemic modeling to traffic prediction and urban
planning, we need quantitative models that can account for the statistical
characteristics of individual human trajectories. Here we use empirical data on
human mobility, captured by mobile phone traces, to show that the predictions
of the CTRW models are in systematic conflict with the empirical results. We
introduce two principles that govern human trajectories, allowing us to build a
statistically self-consistent microscopic model for individual human mobility.
The model not only accounts for the empirically observed scaling laws but also
allows us to analytically predict most of the pertinent scaling exponents
Random walk with barriers: Diffusion restricted by permeable membranes
Restrictions to molecular motion by barriers (membranes) are ubiquitous in
biological tissues, porous media and composite materials. A major challenge is
to characterize the microstructure of a material or an organism
nondestructively using a bulk transport measurement. Here we demonstrate how
the long-range structural correlations introduced by permeable membranes give
rise to distinct features of transport. We consider Brownian motion restricted
by randomly placed and oriented permeable membranes and focus on the
disorder-averaged diffusion propagator using a scattering approach. The
renormalization group solution reveals a scaling behavior of the diffusion
coefficient for large times, with a characteristically slow inverse square root
time dependence. The predicted time dependence of the diffusion coefficient
agrees well with Monte Carlo simulations in two dimensions. Our results can be
used to identify permeable membranes as restrictions to transport in disordered
materials and in biological tissues, and to quantify their permeability and
surface area.Comment: 8 pages, 3 figures; origin of dispersion clarified, refs adde
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