Reconstruction Threshold for the Hardcore Model

In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.Comment: 14 pages, 2 figure

The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective

Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.Comment: 151 pages, 21 figure