Graphical Representations for Ising Systems in External Fields

A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin-Teller) model. Bonds in the graphical representation connect the Ashkin-Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.Comment: 13 pages amste

The Advantage of Foraging Myopically

We study the dynamics of a \emph{myopic} forager that randomly wanders on a lattice in which each site contains one unit of food. Upon encountering a food-containing site, the forager eats all the food at this site with probability $p<1$; otherwise, the food is left undisturbed. When the forager eats, it can wander $\mathcal{S}$ additional steps without food before starving to death. When the forager does not eat, either by not detecting food on a full site or by encountering an empty site, the forager goes hungry and comes one time unit closer to starvation. As the forager wanders, a multiply connected spatial region where food has been consumed---a desert---is created. The forager lifetime depends non-monotonically on its degree of myopia $p$, and at the optimal myopia $p=p^*(\mathcal{S})$, the forager lives much longer than a normal forager that always eats when it encounters food. This optimal lifetime grows as $\mathcal{S}^2/\ln\mathcal{S}$ in one dimension and faster than a power law in $\mathcal{S}$ in two and higher dimensions.Comment: 10 pages, 1o figure

Facilitated diffusion of proteins on chromatin

We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding/unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor, and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.Comment: 4 pages, 4 figures, accepted versio