# Probability and statistics seminar: On 1-independent random graphs

- Date: –11:15
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
- Lecturer: Victor Falgas-Ravry, Umeå universitet
- Organiser: Matematiska institutionen
- Contact person: Xing Shi Cai
- Seminarium

**Abstract: **Let *H* be a connected graph. The p-random graph *H**p* is obtained by including each edge of *H* at random with probability *p*, independently of all the rest. The connectivity properties of *H**p* have been widely studied in random graph theory (in particular when *H=K**n* and *H**p* is the Erdős-Rényi random graph model) and in percolation theory (in particular when *H* is the square integer lattice ℤ2).

In this talk, I will be interested in studying the same connectivity properties but in a different class of random graph models, for which there may be some local dependencies between the edges. Formally, a 1-independent model (1-ipm) on *H* is a probability measure *μ* on the subgraphs of *H* such that in a *μ*-random graph *G**μ*, events supported on disjoint vertex-sets are independent.

Consider a 1-ipm *G**μ* on *H* in which each edge is present with probability at least p. If *H* is finite, what can we say about the probability that *G**μ* is connected? If *H* is infinite, what can we say about the probability that *G**μ* contains an infinite connected component? I will report some recent (modest) progress on these questions and, time allowing, I will discuss some of the many open problems in the area.

Joint work with A. Nicholas Day and Robert Hancock.