# Probability Seminar

# Fall 2014

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

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## Thursday, April 10, Dan Romik UC-Davis

Title: **Connectivity patterns in loop percolation and pipe percolation**

Abstract:

Loop percolation is a random collection of closed cycles in the square lattice Z^2, that is closely related to critical bond percolation. Its "connectivity pattern" is a random noncrossing matching associated with a loop percolation configuration that encodes information about connectivity of endpoints. The same probability measure on noncrossing matchings arises in several different and seemingly unrelated settings, for example in connection with alternating sign matrices, the quantum XXZ spin chain, and another type of percolation model called pipe percolation. In the talk I will describe some of these connections and discuss some results about the study of pipe percolation from the point of view of the theory of interacting particle systems. I will also mention the "rationality phenomenon" which causes the probabilities of certain natural connectivity events to be dyadic rational numbers such as 3/8, 97/512 and 59/1024. The reasons for this are not completely understood and are related to certain algebraic conjectures that I will discuss separately in Friday's talk in the Applied Algebra seminar.

## Thursday, May 1, Antonio Auffinger U Chicago

Title: **Strict Convexity of the Parisi Functional**

Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.

## Thursday, May 8, Steve Goldstein, WID

Title: **Modeling patterns of DNA sequence diversity with Cox Processes**

Abstract: Events in the evolutionary history of a population can leave subtle signals in the patterns of diversity of its DNA sequences. Identifying those signals from the DNA sequences of present-day populations and using them to make inferences about selection is a well-studied and challenging problem.

Next generation sequencing provides an opportunity for making inroads on that problem. In this talk, I will present a novel model for the analysis of sequence diversity data and use the model to motivate analyses of whole-genome sequences from 11 strains of Drosophila pseudoobscura.

The model treats the polymorphic sites along the genome as a realization of a Cox Process, a point process with a random intensity. Within the context of this model, the underlying problem translates to making inferences about the distribution of the intensity function, given the sequence data.

We give a proof of principle, showing that even a simplistic application of the model can quantify differences in diversity between regions with varying recombination rates. We also suggest a number of directions for applying and extending the model. -->