On the phase transition in random simplicial complexes

Seminar
Speaker
Yuval Peled (Hebrew University)
Date
30/11/2014 - 15:30 - 14:00Add to Calendar 2014-11-30 14:00:00 2014-11-30 15:30:00 On the phase transition in random simplicial complexes It is well-known that the G(n,p) model of random graphs undergoes a dramatic change around p=1/n. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant connected component. Several years ago, Linial and Meshulam have introduced the X_d(n,p) model, a probability space of n-vertex d-dimensional simplicial complexes, where X_1(n,p) coincides with G(n,p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from X_d(n,p), and show that it is strictly greater than the threshold of d-collapsibility. In addition, we compute the real Betti numbers, i.e. the dimension of the homology groups, of X_d(n,p) for p=c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d=1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d > 1  the emergence of the giant shadow is a first order phase transition. The talk will contain the necessary toplogical backgorund on simplicial complexes, and will focus on the main idea of the proof: the local weak limit of random simplicial complexes and its role in the analysis of phase transitions. Joint work with Nati Linial.  אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

It is well-known that the G(n,p) model of random graphs undergoes a dramatic change around p=1/n. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant connected component. Several years ago, Linial and Meshulam have introduced the X_d(n,p) model, a probability space of n-vertex d-dimensional simplicial complexes, where X_1(n,p) coincides with G(n,p). Within this model we prove a natural d-dimensional analog of these graph theoretic phenomena. Specifically, we determine the exact threshold for the nonvanishing of the real d-th homology of complexes from X_d(n,p), and show that it is strictly greater than the threshold of d-collapsibility. In addition, we compute the real Betti numbers, i.e. the dimension of the homology groups, of X_d(n,p) for p=c/n. Finally, we establish the emergence of giant shadow at this threshold. (For d=1 a giant shadow and a giant component are equivalent). Unlike the case for graphs, for d > 1  the emergence of the giant shadow is a first order phase transition.

The talk will contain the necessary toplogical backgorund on simplicial complexes, and will focus on the main idea of the proof: the local weak limit of random simplicial complexes and its role in the analysis of phase transitions.

Joint work with Nati Linial. 

Last Updated Date : 27/11/2014