## Student Probability Seminar

Autumn 2017: We studied non-standard random walk models. Papers used include:

Angel, Benjamini, Virág 2003: "Random walks that avoid their past convex hull"

Zerner 2005: "On the speed of a planar random walk avoiding its past convex hull"

Benjamini, Wilson 2003: "Excited random walk"

Lawler 2013: "Intersections of random walks"

Diaconis 1988: "Recent progress on de Finetti's notions of exchangeability"

Pemantle 1988: "Phase transition in reinforced random walk and RWRE on trees"

Pemantle 2007: "A survey of random processes with reinforcement"

Sabot, Zeng 2016: "A random Schrödinger operator associated with the Vertex Reinforced Jump Process on infinite graphs"

Hairer, Labbé 2015: "A simple construction of the parabolic Anderson model on R^2"

König 2016: "The parabolic Anderson model: Random walk in random potential"

Sinai 1982: "The limiting behavior of a one-dimensional random walk in a random medium"

Avena, den Hollander 2016: "Random walks in cooling random environments"

Eischelbacher, König 2008: "Ordered random walks"

Dyson 1962: "A Brownian-Motion Model for the Eigenvalues of a Random Matrix"

Alon, Benjamini, Lubetzky, Sodin 2007: "Non-backtracking random walks mix faster"

Ben Arous, Cerny 2006: "Dynamics of trap models"

October 6

October 13

October 20

October 27

November 3

November 10

November 17

November 24

December 1

December 8

Erik Bates

Andy Tsao

Mark Perlman

Andrea Ottolini

Alex Dunlap

Damian Pavlyshyn

Kevin Yang

Joey Zou

Leila Sloman

Random walks avoiding their past convex hull

Excited random walks

Self-avoiding and loop-erased random walks

Reinforced edge random walk

The parabolic Anderson model

Random walks in cooling random environments

Non-intersecting random walks

< week off >

Non-backtracking random walks

The Bouchaud trap model

Spring 2017: We studied supercritical branching processes and their limiting measures, as well as continuous analogs. Papers used include:

Liu 2000: "On generalized multiplicative cascades"

Franchi 1995: "Chaos multiplicatif: un traitement simple et complet de la fonction de partition"

Hu, Shi 2009: "Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees"

Barral, Rhodes, Vargas 2012: "Limiting laws of supercritical branching random walks"

Berestycki notes: "Topics on branching Brownian motion"

Lalley, Sellke 1987: "A conditional limit theorem for frontier of a branching Brownian motion"

Hamel, Nolen, Roquejoffre, Ryzhik 2013: "A short proof of the logarithmic Bramson correction in Fisher-KPP equations"

Nolen, Roquejoffre, Ryzhik 2016: "Refined long time asymptotics for Fisher-KPP fronts"

Perkins notes: "Super-Brownian motion and critical stochastic spatial systems"

Slade 2002: "Scaling limits and super-Brownian motion"

Lalley, Perkins, Zheng 2014: "A phase transition for measure-valued SIR epidemic processes"

Bhattacharya, Perlman 2017: "Time-inhomogeneous branching processes conditioned on non-extinction"

Diaconis 1977: "Finite forms of de Finetti's theorem on exchangeability"

Diaconis, Freedman 1980: "Finite exchangeable sequences"

Diaconis, Freedman 1987: "A dozen of de Finetti-style results in search of a theory"

April 14

April 21

April 28

May 5

May 12

May 19

May 26

June 2

Erik Bates

Erik Bates

Alex Zhai

Joey Zou

Cole Graham

Alex Dunlap

Mark Perlman

Andrea Ottolini

Introduction to supercritical branching processes

The phase transition for branching random walk

Measures on end spaces after renormalization

Some aspects of branching Brownian motion

Fisher-KPP at large times: exploring the logarithmic Bramson shift

It's a bird! It's a plane! It's super-Brownian motion!

Time-dependent branching processes

Introduction to de Finetti's theorem(s)

Winter 2017: We studied Interacting Particle Systems from Liggett's book.

January 20

January 27

February 10

February 17

February 24

March 3

March 10

March 17

Alex Dunlap

Alex Dunlap

Andy Tsao

Erik Bates

Alex Dunlap

Qian Zhao

Beniada Shabani

Introduction to interacting particle systems

Construction and basic ergodicity properties of some interacting particle systems

Monotonicity and positive correlation methods in interacting particle systems

Examples and applications of dual processes

Wave speed of the contact process & an application to the stochastic Fisher-KPP equation

Introduction to the stochastic Ising model

< week off >

Limiting behavior of Brunet-Derrida particle systems

Autumn 2016: We studied Stein's method, referring to the survey by Ross and notes by Chatterjee.

October 10

October 17

October 24

November 1

November 7

November 14

November 28

December 5

Erik Bates

Paulo Orenstein

Jimmy He

Alex Dunlap

Mark Perlman

Erik Bates

Leila Sloman

Introduction to Stein's method; sums of random variables with sparse dependency graphs

Introduction to exchangeable pairs

Size-bias coupling

Exponential approximation and an application to critical branching processes

Zero-bias coupling and the Lindeberg condition

Geometric approximation with an application to uniform attachment graph model

< week off >

Concentration inequalities via Stein's method

Spring 2016: We followed van Handel's Probability in High Dimension.

April 8

April 15

April 22

April 29

May 6

May 13

Erik Bates

Erik Bates

Subhabrata Sen

Zhou Fan

Paulo Orenstein

Alex Zhai

Chapter 1 and Section 2.1

Sections 2.2 - 2.4

Sections 3.1 - 3.3

Section 3.4

Sections 4.1 and 4.2

Concentration results via Brownian motion

Winter 2016: We followed Dembo-Zeitouni's Large Deviations Techniques and Applications.

January 21

January 28

February 4

February 11

February 18

February 25

March 3

Erik Bates

Zhou Fan

Alex Dunlap

Subhabrata Sen

Erik Bates

Naomi Feldheim

Alex Zhai

Rate functions, Sanov's Theorem

Cramér's Theorem

Gärtner–Ellis Theorem

LDPs for Markov chains and random walks

Gibbs conditioning principle

LDPs on general topological spaces

Introduction to nonlinear large deviations