This paper contains three types of results:the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane,the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane,the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces.In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result
We discuss how information encoded in a template polymer can be stochastically copied into a copy polymer. We consider four different stochastic copy protocols of increasing complexity, inspired by building blocks of the mRNA translation pathway. In the rst protocol, monomer incorporation occurs in a single stochastic transition. We then move to a more elaborate protocol in which an intermediate step can be used for error correction. Finally, we discuss the operating regimes of two kinetic proofreading protocols: one in which proofreading acts from the nal copying step, and one in which it acts from an intermediate step. We review known results for these models and, in some cases, extend them to analyze all possible combinations of energetic and kinetic discrimination. We show that, in each of these protocols, only a limited number of these combinations leads to an improvement of the overall copying accuracy
We present a detailed investigation of the behavior of the nonlinear q-voter model for opinion dynamics. At the mean-field level we derive analytically, for any value of the number q of agents involved in the elementary update, the phase diagram, the exit probability and the consensus time at the transition point. The mean-field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks. We finally discuss the case of random regular networks and compare analytical results with simulations.
The suitability of the generalized Langevin equation (GLE) for a realistic description of the behavior of a system of interacting particles in solution is discussed. This study is focused on the GLE for a system of non-Brownian particles, i.e., the masses and the sizes of the solute particles are similar to those of the bath particles. The random and frictional forces on the atoms of the solute due to their collisions with the solvent atoms are characterized from molecular dynamics simulations of simple dense liquid mixtures. The required effective memory functions, which are dependent on the concentration of solute, are obtained by solving a generalized Volterra equation. The validity of the usual assumptions on the statistical properties of the random forces is carefully analyzed, paying special attention to their Gaussianity. The reliability of stochastic simulations based on the GLE is also discussed.