Related papers: How to count in hierarchical landscapes: a 'full' …
Finding the mean of the total number of stationary points for N-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is…
We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…
Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses…
This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the…
The application of numerical techniques to the study of energy landscapes of large systems relies on sufficient sampling of the stationary points. Since the number of stationary points is believed to grow exponentially with system size, we…
This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple…
The stationary points (SPs) of a potential energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. Unless they are found analytically, there is, however, no efficient method to…
We introduce a model for a sandpile, with N sites, critical height N and each site connected to every other site. It is thus a mean-field model in the spin-glass sense. We find an exact solution for the steady state probability distribution…
The present article is based on a previous one, where a second quantized field theory on the world sheet for summing the planar graphs of phi^3 theory was developed. In this earlier work, the ground state of the model was determined using a…
Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both…
Let $k$ be a finite field extension of the function field $\bfF_p(T)$ and $\bar{k}$ its algebraic closure. We count points in projective space $\Bbb P ^{n-1}(\bar{k})$ with given height and of fixed degree $d$ over the field $k$. If…
Here, we consider a regularized mean-field game model that features a low-order regularization. We prove the existence of solutions with positive density. To do so, we combine a priori estimates with the continuation method. In contrast…
We study models for surface growth with a wetting and a roughening transition using simple and pair mean-field approximations. The simple mean-field equations are solved exactly and they predict the roughening transition and the correct…
We consider the problem of approximating ground states of one-dimensional quantum systems within the two most common variational ansatzes, namely the mean field ansatz and Matrix Product States. We show that both for mean field and for…
We construct an integral model for counting Campana points of bounded height on diagonal hypersurfaces of degree greater than one, and give an asymptotic formula for their number, generalising work by Browning and Yamagishi. The paper also…
The main difficulty that arises in the analysis of most machine learning algorithms is to handle, analytically and numerically, a large number of interacting random variables. In this Ph.D manuscript, we revisit an approach based on the…
We propose an accurate method for counting states of close to zero and positive cosmological constant in the Bousso-Polchinski Landscape. This method is based on simple geometrical considerations on the high-dimensional lattice of quantized…
There has been recent progress on the problem of inferring the structure of interactions in complex networks when they are in stationary states satisfying detailed balance, but little has been done for non-equilibrium systems. Here we…
In this paper we study iterative procedures for stationary equilibria in games with large number of players. Most of learning algorithms for games with continuous action spaces are limited to strict contraction best reply maps in which the…
A mathematically rigorous Hamiltonian formulation for classical and quantum field theories is given. New results include clarifications of the structure of linear fields, and a plausible formulation for nonlinear fields. Many mathematical…