Integrals, Partitions, and Cellular Automata

We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.Comment: Revised version. 28 pages, 2 figure

Gaussian Mean Fields Lattice Gas

We study rigorously a lattice gas version of the Sherrington-Kirckpatrick spin glass model. In discrete optimization literature this problem is known as Unconstrained Binary Quadratic Programming (UBQP) and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present an heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances.Comment: 3 figures, 2 table

05201 Abstracts Collection -- Design and Analysis of Randomized and Approximation Algorithms

From 15.05.05 to 20.05.05, the Dagstuhl Seminar 05201 ``Design and Analysis of Randomized and Approximation Algorithms\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Phase transitions for the cavity approach to the clique problem on random graphs

We give a rigorous proof of two phase transitions for a disordered system designed to find large cliques inside Erdos random graphs. Such a system is associated with a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure