# Difference between revisions of "Probability Seminar"

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− | = | + | = Fall 2019 = |

− | <b>Thursdays in 901 Van Vleck Hall at 2: | + | <b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted. |

− | <b>We usually end for questions at 3: | + | <b>We usually end for questions at 3:20 PM.</b> |

− | If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu. | + | If you would like to sign up for the email list to receive seminar announcements then please send an email to |

+ | [mailto:join-probsem@lists.wisc.edu join-probsem@lists.wisc.edu] | ||

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− | == | + | == September 12, 2019, [https://perso.univ-rennes1.fr/victor.kleptsyn/ Victor Kleptsyn], CNRS and University of Rennes 1 == |

− | + | '''Furstenberg theorem: now with a parameter!''' | |

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− | + | The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. | |

+ | It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes. | ||

+ | Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions. | ||

− | + | == September 19, 2019, [http://math.columbia.edu/~xuanw Xuan Wu], Columbia University== | |

− | + | '''A Gibbs resampling method for discrete log-gamma line ensemble.''' | |

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− | + | In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits. | |

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− | == | + | == October 10, 2019, NO SEMINAR - [https://sites.math.northwestern.edu/mwp/ Midwest Probability Colloquium] == |

− | + | == October 17, 2019, [https://www.usna.edu/Users/math/hottovy/index.php Scott Hottovy], USNA == | |

+ | ''' Simplified dynamics for noisy systems with delays.''' | ||

− | + | Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev. | |

− | + | == October 24, 2019, [https://math.temple.edu/~brider/ Brian Rider], Temple University == | |

− | + | '''A general beta crossover ensemble''' | |

− | In | + | I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica). |

− | == | + | == October 31, 2019, [http://math.mit.edu/~elmos/ Elchanan Mossel], MIT == |

− | == | + | == November 7, 2019, [https://people.kth.se/~tobergg/ Tomas Berggren], KTH Stockholm == |

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− | + | == November 14, 2019, [https://math.mit.edu/directory/profile.php?pid=2076 Benjamin Landon], MIT == | |

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− | == | + | == November 21, 2019, Tung Nguyen, UW Madison == |

− | + | == November 28, 2019, Thanksgiving (no seminar) == | |

− | + | == December 5, 2019, Vadim Gorin, UW Madison == | |

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− | == | + | == <span style="color:red"> Wednesday, February 6 at 4:00pm in Van Vleck 911</span> , [https://lc-tsai.github.io/ Li-Cheng Tsai], [https://www.columbia.edu/ Columbia University] == |

− | Title: ''' | + | Title: '''When particle systems meet PDEs''' |

− | + | Abstract: Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.. | |

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− | + | == <span style="color:red">'''Tuesday''' </span>, May 7, Van Vleck 901, 2:25pm, Duncan Dauvergne (Toronto) == | |

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− | + | <div style="width:250px;height:50px;border:5px solid black"> | |

− | + | <b><span style="color:red">  Please note the unusual day. | |

− | + |   </span></b> | |

− | + | </div> | |

− | + | Title: '''The directed landscape''' | |

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− | Title: ''' | ||

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+ | Abstract: I will describe the construction of the full scaling limit of (Brownian) last passage percolation: the directed landscape. The directed landscape can be thought of as a random scale-invariant `directed' metric on the plane, and last passage paths converge to directed geodesics in this metric. The directed landscape is expected to be a universal scaling limit for general last passage and random growth models (i.e. TASEP, the KPZ equation, the longest increasing subsequence in a random permutation). Joint work with Janosch Ormann and Balint Virag. | ||

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## Latest revision as of 19:04, 10 October 2019

# Fall 2019

**Thursdays in 901 Van Vleck Hall at 2:30 PM**, unless otherwise noted.
**We usually end for questions at 3:20 PM.**

If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu

## September 12, 2019, Victor Kleptsyn, CNRS and University of Rennes 1

**Furstenberg theorem: now with a parameter!**

The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices; their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes. Our results are related to the Anderson localization in dimension one, providing a purely dynamical viewpoint on its proof. I will also speak about some generalizations and related open questions.

## September 19, 2019, Xuan Wu, Columbia University

**A Gibbs resampling method for discrete log-gamma line ensemble.**

In this talk we will construct the discrete log-gamma line ensemble, which is assocaited with inverse gamma polymer model. This log-gamma line ensemble enjoys a random walk Gibbs resampling invariance that follows from the integrable nature of the inverse gamma polymer model via geometric RSK correspondance. By exploiting such resampling invariance, we show the tightness of this log-gamma line ensemble under weak noise scaling. Furthermore, a Gibbs property, as enjoyed by KPZ line ensemble, holds for all subsequential limits.

## October 10, 2019, NO SEMINAR - Midwest Probability Colloquium

## October 17, 2019, Scott Hottovy, USNA

** Simplified dynamics for noisy systems with delays.**

Many biological and physical systems include some type of random noise with a temporal delay. For example, many sperm cells travel in a random motion where their velocity changes according to a chemical signal. This chemotaxis is transmitted through a delay in the system. That is, the sperm notices chemical gradients after a certain time has elapsed. In this case, the delay causes the sperm to aggregate around the egg. In this talk I will consider a general stochastic differential delay equation (SDDE) with state-dependent colored noises and derive its limit as the time delays and the correlation times of the noises go to zero. The analysis leads to a much simpler Stochastic Differential Equation to study. The work is motivated by an experiment involving an electrical circuit with noisy, delayed feedback. The main methods used in the proof are a theorem about convergence of solutions of stochastic differential equations by Kurtz and Protter and a maximal inequality for sums of a stationary sequence of random variables by Peligrad and Utev.

## October 24, 2019, Brian Rider, Temple University

**A general beta crossover ensemble**

I'll describe an operator limit for a family of general beta ensembles which exhibit a double-scaling. In particular, a free parameter in the system provides for a crossover between the more well-known "soft" and "hard" edge point processes. This new limit operator takes as input the Riccati diffusion associated with the Stochastic Airy Operator. I like to suggest that this hints at a hierarchy of random operators analogous to the Painlevé hierarchy observed at the level of correlation functions for double-scaling ensembles most widely studied at beta = 2. Full disclosure: the result remains partially conjectural due to an unresolved uniqueness question, but I’ll provide lots of evidence to convince you we have the right answer. Joint work with Jose Ramírez (Univ. Costa Rica).