Abstract for the talk at 12.02.2009 (15:15 h)

Pietro Majer (Università di Pisa, Italy)

Infinite paths in random graphs (some intersection lemmas in measure theory)

Joint work with A. Berarducci and M. Novaga

Some existence problems concerning subsequences with special properties, in a context of dynamical systems, ask for special intersection lemmas in measure theory. The archetype of this situation is the recurrence theorem of Poincare', and the Borel-Cantelli lemma. I will discuss some of these intersection problem. For instance, in the simplest form, we have: PROBLEM. Let be a double sequence of masurable subsets in a probability space Omega, with indices over all pairs i0. Is there an increasing sequence of numbers , we look for sharp estimates on the probability of percolation, that is, for instance, in the above mentioned example, estimates on the measure of the event: x : there exists a sequence i_0<.. such that x X_i_1,i_2 X_i_2,i_3 X_i_3,i_4..... The computation is made possible after a reduction to a suitable variational problem. While doing this reduction, one is naturally lead to employ various mathematical theories : Ramsey theory; de Finetti's exchangeability theory and its more recent extensions (Aldous-Hoover Kallenberg); transfinite ordinals; elementary ergodic theory.