### Yale University

Department of Statistics

Seminar

#### Monday, October 16, 1995

Professor Joel Spencer

Courant Institute, New York University

#### "The Percolation of the Random Graph"

Seminar to be held in Room 107, 24 Hillhouse Ave at 4:15 pm

Paul Erdos and Alfred Renyi showed thirty years ago that
the random graph G(n,p) undergoes percolation (what they called
The Double Jump) at p=1/n. Today we know how to slow down the
process, using the parametrization np-1=an^{-1/3}.
We approach this through classic percolation,
considering a branching process with mean near one. The infinite
and finite components have analogs in our
finite, asymptotic, case. As a surprising Corollary we get
asymptotic enumeration of unicylic (bicyclic, etc.) graphs in
terms of moments of the area under a Brownian Bridge.