Related papers: Optimization of random cost functions and statisti…
This thesis presents original results in two domains of disordered statistical physics: logarithmic correlated Random Energy Models (logREMs), and localization transitions in long-range random matrices. In the first part devoted to logREMs,…
We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect…
The article reviews recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples. Various spatial geometries are considered, with emphasis on…
In these lectures I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to…
We consider a random Hamiltonian $H:\Sigma\to\mathbb R$ defined on a compact space $\Sigma$ that admits a transitive action by a compact group $\mathcal G$. When the law of $H$ is $\mathcal G$-invariant, we show its expected free energy…
Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N-1)- dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the "trust…
Spin glass theory studies the structure of sublevel sets and minima (or near-minima) of certain classes of random functions in high dimension. Near-minima of random functions also play an important role in high-dimensional statistics and…
LECTURE GIVEN AT TH2002. Given a set of Boolean variables, and some constraints between them, is it possible to find a configuration of the variables which satisfies all constraints? This problem, which is at the heart of combinatorial…
In string theory with flux compactifications, anthropic selection for structure formation from a discretuum of vacuum energy values provides at present our only understanding of the tiny yet positive value of the cosmological constant. We…
Distributing points on a (possibly high-dimensional) sphere with minimal energy is a long-standing problem in and outside the field of mathematics. This paper considers a novel energy function that arises naturally from statistics and…
We consider a finite horizon stochastic optimal control problem for nearest-neighbor random walk $\{X_i\}$ on the set of integers. The cost function is the expectation of exponential of the path sum of a random stationary and ergodic…
We analyse several constructions of random point sets on the sphere $\mathbb{S}^{3}\subset\mathbb{R}^4$ evaluating and comparing them through their discrete logarithmic energy: \begin{equation*} E_0(\omega_N) = \sum_{\substack{i, j=1\\ i…
Previous large $N$ calculations are combined with numerical work at $N=4$ to show that the Minimal Standard Model will describe physics to an accuracy of a few percent up to energies of the order 2 to 4 times the Higgs mass, $M_H$, only if…
This paper focuses on finding approximate solutions to stochastic optimal control problems with control domains being not necessarily convex, where the state trajectory is subject to controlled stochastic differential equations. The…
Optimization problems on probability measures in $\mathbb{R}^d$ are considered where the cost functional involves multi-marginal optimal transport. In a model of $N$ interacting particles, like in Density Functional Theory, the interaction…
In this paper we present a new point of view on the mathematical foundations of statistical physics of infinite volume systems. This viewpoint is based on the newly introduced notions of transition energy function, transition energy field…
It is shown {\it in detail how} the ground-state self-energy $\Sigma(k,\omega)$ of the spin-unpolarized uniform electron gas (with the density parameter $r_s$) in its high-density limit $r_s\to 0 $ determines: the momentum distribution…
These notes are based on lectures given at the Les Houches Summer School in 2011, which was centered on the general topic "Theoretical Physics to face the challenge of LHC". In these lectures I reviewed a number of topics in the field of…
Energy saving is becoming an important issue in the design and use of computer networks. In this work we propose a problem that considers the use of rate adaptation as the energy saving strategy in networks. The problem is modeled as an…
In physics, there is a scalar function called the action which behaves like a cost function. When minimized, it yields the "path of least action" which represents the path a physical system will take through space and time. This function is…