We demonstrate that the fraction of pattern sets that can be stored in singleand hidden-layer perceptrons exhibits finite size scaling. This feature allows one to estimate the critical storage capacity ac from simulations of relatively small systems. We illustrate this approach by determining ac , together with the finite size scaling exponent n, for storing Gaussian patterns in committee and p...

In this letter we investigate the finite size scaling effect on SLE(κ, ρ) and boundary conformal field theories and find the effect of fixing some boundary conditions on the free energy per length of SLE(κ, ρ). As an application, we will derive the entanglement entropy of quantum systems in critical regime in presence of boundary operators.

Singularities of thermodynamic quantities in equilibrium critical phenomena are characterized by well defined critical exponents. The set of critical exponents determine the universality class of a system in the thermodynamic limit. In numerical simulations, the results are very often limited by the finite system size L. There is rounding and shifting of critical singularities in finite systems...

The exact solution of directed self-avoiding walks confined to a slit of finite width and interacting with the walls of the slit via an attractive potential has been recently calculated. The walks can be considered to model the polymerinduced steric stabilization and sensitized flocculation of colloidal dispersions. The large-width asymptotics led to a phase diagram different to that of a polym...

The present review is devoted to the problems of finite-size scaling due to the presence of long-range interaction decaying at large distance as 1/rd+σ, where d is the spatial dimension and the long-range parameter σ > 0. Classical and quantum systems are considered. 1. Finite-size systems and critical phenomena A common wisdom is that the singularities in the thermodynamic functions at a criti...

We present an analysis of finite-size effects in jammed packings of N soft, frictionless spheres at zero temperature. There is a 1/N correction to the discrete jump in the contact number at the transition so that jammed packings exist only above isostaticity. As a result, the canonical power-law scalings of the contact number and elastic moduli break down at low pressure. These quantities exhib...

According to the eigenstate thermalization hypothesis (ETH), even isolated quantum systems can thermalize because the eigenstate-to-eigenstate fluctuations of typical observables vanish in the limit of large systems. Of course, isolated systems are by nature finite and the main way of computing such quantities is through numerical evaluation for finite-size systems. Therefore, the finite-size s...

Abstract. Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time N , and N is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their exactly solvable character, while displ...

We study numerically the properties of the Bayesian perceptron through a gradient descent on the optimal cost function. The theoretical distribution of stabilities is deduced. It predicts that the optimal generalizer lies close to the boundary of the space of ~error-free! solutions. The numerical simulations are in good agreement with the theoretical distribution. The extrapolation of the gener...