# Probability Seminar

# Spring 2015

**Thursdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.

**
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**

## Thursday, January 15, Miklos Racz, UC-Berkeley Stats

Title: TBA

## Monday, December 1, Joe Neeman, UT-Austin, 4pm, Room B239 Van Vleck Hall

Please note the unusual time and room.

Title: **Some phase transitions in the stochastic block model**

Abstract: The stochastic block model is a random graph model that was originally 30 years ago to study community detection in networks. To generate a random graph from this model, begin with two classes of vertices and then connect each pair of vertices independently at random, with probability p if they are in the same class and probability q otherwise. Some questions come to mind: can we reconstruct the classes if we only observe the graph? What if we only want to partially reconstruct the classes? How different is this model from an Erdos-Renyi graph anyway? The answers to these questions depend on p and q, and we will say exactly how.

## Thursday, December 4, Arjun Krishnan, Fields Institute

Title: **Variational formula for the time-constant of first-passage percolation**

Abstract: Consider first-passage percolation with positive, stationary-ergodic weights on the square lattice in d-dimensions. Let [math]T(x)[/math] be the first-passage time from the origin to [math]x[/math] in [math]Z^d[/math]. The convergence of [math]T([nx])/n[/math] to the time constant as [math]n[/math] tends to infinity is a consequence of the subadditive ergodic theorem. This convergence can be viewed as a problem of homogenization for a discrete Hamilton-Jacobi-Bellman (HJB) equation. By borrowing several tools from the continuum theory of stochastic homogenization for HJB equations, we derive an exact variational formula (duality principle) for the time-constant. Under a symmetry assumption, we will use the variational formula to construct an explicit iteration that produces the limit shape.

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