# Duke Probability Seminar

### A seminar for the probability community at Duke, both in and outside of the Mathematics Department.

#### More information on the Duke Probability community can be found on the Probability: Theory and Applications Page

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Upcoming Seminars:
• Thursday, September 20, 2018, 3:15pm, 119 Physics, Probability Seminar
Stable coexistence of savannah and forest in a spatial model
Carla Staver

The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

• Thursday, September 27, 2018, 3:15pm, 119 Physics, Probability Seminar
Mean-field diffusions in stochastic spatial death-birth models.
Yu-ting Chen (U of Tennessee, Knoxville)

In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.

• Thursday, October 4, 2018, 3:15pm, 119 Physics, Probability Seminar
Pattern-avoiding permutations and Dyson Brownian motion
Erik Slivken (Paris Diderot)

Let $$S_n$$ denote the set of permutations of length $$n$$. For a permutation $$\tau \in S_n$$ we say $$\tau$$ contains a pattern $$\sigma\in S_k$$ if there is a subsequence $$i_1 < \cdots < i_k$$ such that $$\tau_{i_1} \cdots \tau_{i_k}$$ has the the same relative order of $$\sigma$$. If $$\tau$$ contains no pattern $$\sigma$$, we say that $$\tau$$ avoids $$\sigma$$. We denote the set of $$\sigma$$-avoiding permutations of length $$n$$ by $$S_n(\sigma)$$. Recently, there has been a number of results that help describe the geometric properties of a uniformly random element in $$S_n(\sigma)$$. Many of these geometric properties are related to well-studied random objects that appear in other settings. For example, if $$\sigma \in S_3$$, we have that a permutation chosen uniformly in $$S_n(\sigma)$$ converges, in some appropriate sense, to Brownian excursion. Furthermore for $$\sigma = 123,312$$ or$$231$$, we can describe properties like the number and location of fixed points in terms of Brownian excursion. Larger patterns are much more difficult to understand. Currently even the simplest question, enumeration, is unknown for the pattern $$\sigma = 4231$$. However, for the monotone decreasing pattern $$\sigma= (d+1)d\cdots 21$$, $$S_n(\sigma)$$ can be coupled with a random walk in a cone that, in some appropriate sense, converges to a traceless Dyson Brownian motion.

• Thursday, October 11, 2018, 3:15pm, 119 Physics, Probability Seminar
Asymptotic behavior of the homology of random polyominoes
Erika Berenice Roldan Roa (Ohio State)

In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.

• Thursday, October 18, 2018, 3:15pm, 119 Physics, Probability Seminar
title
Lisa Hartung (Courant Institute)

abstract

• Thursday, November 8, 2018, 3:15pm, 119 Physics, Probability Seminar
title
Pascal Maillard (CRM (Montréal) and Université Paris-Sud)

abstract

• Thursday, March 28, 2019, 3:15pm, 119 Physics, Probability Seminar
TBA
Kavita Ramanan (Brown)

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