Groningen Stochastics Seminar

If you are interested in giving a talk, please send an email with the title and abstract to b.t.hansen@rug.nl.

Up Next

September 19, 2018 Room 5173.0045 (Linnaeusborg) 15:30 - 16:30 | Dieter Mitsche
(Nice) | k-regular subgraphs near the k-core threshold of a random graph |
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Future Talks

October 10, 2018 Room 5173.0045 (Linnaeusborg) 15:00 - 16:00 | Clara Stegehuis
(Eindhoven) | Optimizing constrained and unconstrained network structures |
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November 7, 2018 Room 5161.0289 14:00 - 15:00 | Alessandra Cipriani
(Delft) | TBA? |

November 21, 2018 Room TBA 15:00 - 16:00 | Guus Regts
(Amsterdam) | TBA? |

November 26, 2018 Room TBA 15:00 - 16:00 | Nelly Litvak
(Eindhoven/Twente) | TBA? |

Recent Past

June 20, 2018 Room 5161.0222 11:00 - 12:00 | Bernardo N. B. de Lima
(Federal University of Minas Gerais) | Truncated long-range percolation on oriented graphs |
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June 18, 2018 Room 5161.0165 15:00 - 16:00 | Moritz Schauer
(Leiden) | A generator approach to stochastic monotonicity and propagation of order |

April 23, 2018 Room 5161.0293 16:00 - 17:00 | Evgeny Verbitskiy
(Leiden) | Solvable models from the ergodic point of view |

March 15, 2018 Room 5115.0020 11:00-12:00 | Rodrigo Ribeiro
(IMPA) | Building your path to escape from home |

February 16, 2018 Room 5161.0222 15:00-16:00 | Renato Santos
(Weierstrass Institute) | Random walk on random walks |

February 1, 2018 Room 5161.0293 16:00 - 17:00 | Conrado Costa
(Leiden) | Law of large numbers and LDP for the Random Walk in the Cooling random Environment |

January 24, 2018 Room 5161.0222 16:00 - 17:00 | Daniel Valesin
(Groningen) | Monotonicity for multi-range percolation on oriented trees |

Abstracts

**Title:** Optimizing constrained and unconstrained network structures

**Speaker:** Clara Stegehuis

**Abstract:**
Subgraphs contain important information about network structures and their functions. We investigate the presence of subgraphs in inhomogeneous random graphs with infinite-variance degrees. We introduce an optimization problem which identifies the dominant structure of any given subgraph. The unique optimizer describes the degrees of the vertices that together span the most likely subgraph and allows us count and characterize all subgraphs.

We then show that this optimization problem easily extends to other network structures, such as clustering, which expresses the probability that two neighbors of a vertex are connected. The optimization problem is able to find the behavior of clustering in a wide class of random graph models.

**Title:** k-regular subgraphs near the k-core threshold of a random graph

**Speaker:** Dieter Mitsche

**Abstract:**
We prove that $G_{n,p}$ whp has a k-regular subgraph if $c$ is at least $e^{-\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. an non-empty k-core. In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size $e^{-\Theta(k)}$.

Joint work with Mike Molloy and Pawel Pralat.

<p>We prove that <span class="math"><em>G</em><sub><em>n</em>, <em>p</em></sub></span> whp has a k-regular subgraph if <span class="math"><em>c</em></span> is at least <span class="math"><em>e</em><sup> − Θ(<em>k</em>)</sup></span> above the threshold for the appearance of a subgraph with minimum degree at least k; i.e. an non-empty k-core. In particular, this pins down the threshold for the appearance of a k-regular subgraph to a window of size <span class="math"><em>e</em><sup> − Θ(<em>k</em>)</sup></span>.</p>

**Title:** Truncated long-range percolation on oriented graphs

**Speaker:** Bernardo N. B. de Lima

**Abstract:**
We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the context of a long-range contact process.

Joint work with A. Alves, A. van Enter, M. Hilário and D. Valesin

**Title:** A generator approach to stochastic monotonicity and propagation of order

**Speaker:** Moritz Schauer

**Abstract:**
We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying Φ f ≥ 0 for some linear operator Φ.

We introduce a new functional analytic technique based on the generator A of a Markov process and its resolvent. We show that the existence of an operator B with positive resolvent such that Φ A - B Φ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order that can be applied in a wide range of settings including various orders for diffusion processes with or without boundary conditions and orders for discrete interacting particle systems.

Joint work with Richard Kraaij

**Title:** Solvable models from the ergodic point of view

**Speaker:** Evgeny Verbitskiy

**Abstract:** In this review talk, I will discuss a link between solvable models of statistical mechanics (dimers, spanning trees, sandpiles) and algebraic dynamical systems. Even though the question about the existence of such a link was raised almost two decades ago, this problem remained largely inaccessible. The development of the theory of symbolic covers of algebraic dynamical systems has only recently provided a suitable framework.

**Title:** Building your path to escape from home

**Speaker:** Rodrigo Ribeiro

**Abstract:** We consider a random walker in a dynamic random environment
given by a system of independent simple symmetric random walks.
We will describe some perturbative results that can be obtained via multi-scale analysis,
including regimes of high density, low density and large drift on particles.
Based on joint works with Oriane Blondel, Marcelo Hilário, Frank den Hollander,
Vladas Sidoravicius and Augusto Teixeira.

**Title:** Random walk on random walks

**Speaker:** Renato Santos

**Abstract:** In this talk we introduce one of the simplest conceivable model of a random walk that can modify its domain. The model works as follows: before every walker step, with probability p a new leaf is added to the vertex currently occupied by the walker. We discuss questions related to the walker such as the speed it moves away from its initial position and how its neighborhood may looks like asymptotically. This is a joint work with D. Figueiredo, G. Iacobellu, R. Olivera and B. Reed.

**Title:** Law of large numbers and LDP for the Random Walk in the Cooling random Environment

**Speaker:** Conrado Costa

**Abstract:** In this talk we discuss 2 main new results: the Strong Law of Large Numbers and a quenched Large deviation principle for the Random Walk in the Cooling random Environment introduced by Avena, den Hollander in 2016. For the proof of these two results, we present two auxiliary results. The first, an ergodic theorem for cooling random variables and the second, a concentration result for the cumulant generating function of the static Random Walk in the random environment.

**Title:** Monotonicity for multi-range percolation on oriented trees

**Speaker:** Daniel Valesin

**Abstract:** Percolation theory contains a variety of monotonicity-related conjectures which are easy and natural to state but challenging to prove. In this talk, we will survey a few of these problems. We will then specialize to the setting of Bernoulli percolation on multi-range oriented trees. The trees we consider are oriented regular trees where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there is percolation. We also show that this curve decreases with respect to the length of the long bonds. Joint work with Bernardo N. B. de Lima and Leonardo T. Rolla.