Program:
March 20
8:30-9:15 Registration
9:15-9:30 Opening
09:30-10:30 I. Benjamini, Comments and problems regarding large graphs.
10:30-11:00 Coffee break
11:00-12:00 N. Curien, Subdiffusivity of random walks on random planar maps, via stationarity.
12:00-14:00 Lunch break
14:00-15:00 C. Bordenave, Entropic inequalities for unimodular networks.
15:00-16:00 S. Martineau, Strict monotonicity of percolation thresholds under covering maps.
16:00-16:30 Coffee break
16:30-17:30 M. Lelarge, Spectral embedding for graph classification.
March 21
09:00-10:00 G. Last, A stable marriage between order and disorder.
10:00-11:00 M. Abert, Point processes, cost and the growth of rank for locally compact groups.
11:00-11:30 Coffee break
11:30-12:30 M.O. Haji-Mirsadeghi, Eternal family trees and dynamics on unimodular random graphs.
12:30-14:30 Lunch break
14:30-15:30 P. Brémaud, Sampling cluster point processes: a review.
15:30-16:30 J. Salez, Emergence of extended states at zero in the spectrum of sparse random graphs.
16:30-17:00 Coffee break
17:00-18:00 V. Anantharam, A notion of entropy for limits of sparse marked graphs.
19:00-21:00 Conference dinner
March 22
09:30-10:30 H. Thorisson, On the modified Palm version.
10:30-11:00 Coffee break
11:00-12:00 L. Decreusefond, Stein-Malliavin method for discrete alpha stable point processes.
12:00-14:00 Lunch break
14:00-15:00 Y. Dhandapani, Central Limit theorem for quasi-local statistics of point processes with fast decay of correlations.
15:00-16:00 A. Khezeli, On the notion of dimension of unimodular discrete spaces.
16:00-16:30 Coffee break
16:30-17:30 D. Coupier, Absence of percolation for Poisson outdegree-one graphs.
Abstracts and Video Links:
- Miklós Abért: Point processes, cost and the growth of rank for locally compact groups. Video
The cost of a vertex transitive graph is the infimum of the expected degree of an invariant random wiring of the graph. Similarly, one can define the cost of a point process on a homogeneous space, as the infimum of the average degree of a factor wiring on its points. It turns out that the cost of a Poisson process is maximal among point processes of the same density, by proving that all free processes weakly contain the Poisson. The cost is related to the growth of the minimal number of generators of lattices in Lie groups. We expect that for semisimple Lie groups, the minimal number of generators is sublinear in the volume except for SL(2,R). We outline partial results in this direction and pose some open problems. One of them is to compute the cost of the Poisson on hyperbolic 3-space: solving this would lead to the solution of a 40 year old problem on Heegaard genus of hyperbolic 3-manifolds. This is joint work with Samuel Mellick.
- Venkat Anantharam: A notion of entropy for limits of sparse marked graphs. Video
Bordenave and Caputo (2014) defined a notion of entropy for probability distributions on rooted graphs with finite expected degree at the root. When such a probability distribution 
- Itai Benjamini: Comments and problems regarding large graphs. Video
We will discuss a couple of results and questions regarding the structure of large graphs. These include vertex transitive graphs, expanders and random graphs.
- Charles Bordenave: Entropic inequalities for unimodular networks. Video
- Pierre Brémaud: Sampling cluster point processes: a review. Video
The theme of this talk is the sampling of cluster and iterated cluster point processes. It is partially a review, mainly of the Brix–Kendall exact sampling method for cluster point processes and its adaptation by Moller and Rasmussen to Hawkes branching point processes on the real line with light-tail fertility rate. A formal proof via Laplace transforms of the validity of the method in terms of general clusters that are not necessarily point processes fits this purpose and allows to include the exact sampling of Boolean models. The main novel aspect of this review is the extension of the above sampling methods to non-Poissonian germ point processes.
- David Coupier: Absence of percolation for Poisson outdegree-one graphs. Video
A Poisson outdegree-one graph is a directed graph based on a marked Poisson point process such that each vertex has only one outgoing edge. We state the absence of percolation for such graphs satisfying two assumptions. The Shield assumption roughly says that the graph is locally determined with possible random horizons. The Loop assumption ensures that any forward branch merges on a loop provided that the Poisson point process is augmented with a finite collection of well-chosen points. This result allows to solve a conjecture by D. Daley, S. Ebert and G. Last on the absence of percolation for the “line-segment model”. In this planar model, a segment is growing from any point of the Poisson process and stops its growth whenever it hits another segment. The random directions are picked independently and uniformly on the unit sphere. This is a joint work with D. Dereudre and S. Le Stum.
- Nicolas Curien: Subdiffusivity of random walks on random planar maps, via stationarity. Video
Random planar maps have been the subject of numerous studies over the last years. They are instance of stationary and reversible random planar maps exhibiting a non-conventional geometry at large scale. Because of their “fractal” geometry, the simple random walk on these random graphs is believed to be subdiffusive, i.e. it displaces slowler than in the regular grid case. We will propose an approach to such results strongly based on the stationary of these random graphs, i.e. the fact that their distributions is invariant under rerooting along the simple random walk. Based on joint work with Cyril Marzouk.
- Laurent Decreusefond: Stein-Malliavin method for discrete alpha stable point processes. Video
The notion of discrete alpha-stable point processes generalizes to point processes the notion of stable distribution. It has been introduced and studied by Davydov, Molchanov and Zuyev a few years ago. Their stability property leaves a large degree of variability in the choice of their driving characteristics but enforces a rich mathematical structure. We show how to build a Dirichlet-Malliavin structure for these processes and we apply this framework to several limit theorems. Some known results for Poisson point processes appear as corollaries of the present theorems.
- Yogeshwaran Dhandapani: Central Limit theorem for quasi-local statistics of point processes with fast decay of correlations. Video
We shall consider Euclidean stationary point processes which have fast decay of correlations i.e., their correlation functions factorize upto an additive error decaying exponentially in the separation distance. By a quasi-local statistic of the point process, we refer to statistics that can be expressed as sum of contributions from the points and the contribution of every point being determined by a random ball around the point whose radius has an exponential tail. There are many well-known point processes and statistics that satisfy these conditions and I will present a generic central limit theorem for the same assuming some ‘verifiable’ moment conditions and a ‘non-trivial’ variance lower bound. This is based on a joint work with B. Blaszczyszyn and J. E. Yukich. We will then discuss a similar generic central limit theorems for point processes on discrete Cayley graphs with polynomial growth. These results apply for quasi-local statistics of many off-critical spin models on such graphs. For example, the off-critical Ising model and level sets of the Massive Gaussian free field. This is joint work with T. R. Reddy and S. Vadlamani. In view of these results and more results on asymptotic normality for statistics of other random graph models, one is lead to ask the same question about local or quasi-local statistics on unimodular random graphs.
- Mir-Omid Haji-Mirsadeghi: Eternal family trees and dynamics on unimodular random graphs. Video
This talk is centered on covariant dynamics on unimodular random graphs and random networks (marked graphs), namely maps from the set of vertices to itself which are preserved by graph or network isomorphisms. Such dynamics are referred to as vertex-shifts here. These dynamics have point-shifts on point processes as a subclass. First we give a classification of vertex-shifts on unimodular random networks. Each such vertex-shift partitions the vertices into a collection of connected components and foils. The latter are discrete analogues the stable manifold of the dynamics. The classification is based on the cardinality of the connected components and foils. Up to an event of zero probability, there are three classes of foliations in a connected component: F/F (with finitely many finite foils), I/F (infinitely many finite foils), and I/I (infinitely many infinite foils). In the especial case of point-shifts on stationary point processes the notion of relative intensity can be defined. This notion formalizes the intuition of invariance of dimension between consecutive foils and it is the key element to prove this result for the Hausdorff unimodular dimension of foils. An infinite connected component of the graph of a vertex-shift on a random network forms an infinite tree with one selected end which is referred to as an Eternal Family Tree. Such trees can be seen as stochastic extensions of branching processes. Unimodular Eternal Family Trees can be seen as extensions of critical branching processes. The class of offspring-invariant Eternal Family Trees, allows one to analyze dynamics on networks which are not necessarily unimodular. These can be seen as extensions of not necessarily critical branching processes. Several construction techniques of Eternal Family Trees are proposed, like the joining of trees or moving the root to a far descendant.
- Ali Khezeli: On the notion of dimension of unimodular discrete spaces. Video
In this talk we will define notions of dimension for unimodular random graphs and point-stationary point processes. These notions are in spirit similar to the Minkowski dimension and the Hausdorff dimension. The key point in the definitions is the use of the mass transport principle which is used indispensably and distinguishes this view point from the previous notions which are defined in the literature. The connections of these definitions to volume growth and other notions of dimension are also discussed, which provide a toolset for calculating the dimension. Discrete analogues of several theorems regarding the dimension of continuum spaces are presented; e.g., the mass distribution principle, Billingsley’s lemma, Frostman’s lemma, and the max-flow min-cut theorem. In addition, the notion of unimodular discrete spaces is introduced which is a common generalization of unimodular random graphs and point-stationary point processes. The dimension of several examples of such spaces will be studied. Different methods for finding upper bounds and lower bounds on the dimension will also be presented and illustrated through these examples.
- Guenter Last: A stable marriage between order and disorder. Video
Stable matchings were introduced in a seminal paper by Gale and Shapley (1962) and play an important role in economics. Following closely Holroyd, Pemantle, Peres and Schramm (2009), we shall first discuss a few basic properties of stable matchings between two discrete point sets (resp. point processes) in Euclidean space, where the points prefer to be close to each other. For comparison we also discuss stable transports from Lebesgue measure to point processes. In the second part of the talk we consider a stable matching between the cubic lattice and a stationary Poisson process (or a determinantal point process) with higher intensity. The matched points form a stationary and ergodic (under lattice shifts) point process with unit intensity that has many remarkable properties. If the intensity of the Poisson process is close to one, then it very much resembles a Poisson process, while for large intensities it approaches the lattice. Moreover, the matched points form a hyperuniform and number rigid point process, in sharp contrast to a Poisson process. Still the pair correlation decays exponentially fast. The talk is based on joint work with M. Klatt and D. Yogeshwaran.
- Marc Lelarge: Spectral embedding for graph classification. Video
Learning on graphs requires a graph feature representation able to discriminate among different graphs while being amenable to fast computation. The graph isomorphism problem tells us that no fast representation of graphs is known if we require the representation to be both invariant to nodes permutation and able to discriminate two non-isomorphic graphs. Most graph representations explored so far require to be invariant. We explore new graph representations by relaxing this constraint. We present a generic embedding of graphs relying on spectral graph theory called Spectral Graph Embedding (SGE). We show that for a large family of graphs, our embedding is still invariant. To evaluate the quality and utility of our SGE, we apply them to the graph classification problem.
- Sebastien Martineau: Strict monotonicity of percolation thresholds under covering maps. Video
Percolation is a model for propagation in porous media that as introduced in 1957 by Broadbent and Hammersley. An infinite graph G models the geometry of the situation and a parameter p embodies its porosity: percolation consists in keeping independently each edge with probability p, erasing it otherwise, and looking at the infinite connected components of the resulting graph. It turns out that there is a critical porosity: for smaller porosities, all components are finite almost surely, while for larger ones, there is almost surely at least one infinite component. How does this critical porosity depend on the underlying graph? This is a broad question, that also has connections with the behaviour at the critical point. In this talk, we will consider this question in the following perspective: we will prove that, under reasonable conditions, quotienting a graph strictly increases it critical porosity. This is joint work with Franco Severo.
- Justin Salez: Emergence of extended states at zero in the spectrum of sparse random graphs. Video
We confirm the long-standing prediction that c=e≈2.718 is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős-Renyi random graph with average degree c. This is achieved by a detailed second-order analysis of the resolvent (A−z)−1 near the singular point z=0, where A is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring c. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution. Joint work with Simon Coste.
- Hermann Thorisson: On the modified Palm version. Video
The Palm version of a stationary random measure is an important tool in probability. It is however not well known that there are in fact two Palm versions, with related but different interpretations. For lack of better terms, call the well known version standard and the less known version modified. In this talk we shall focus on the modified Palm version and its interpretation. The concepts of shift-coupling and mass-stationarity will play a key role.
