Analysis and Probability Seminar

The seminar is joint for the division of Analysis and Probability and its main themes are Mathematical Physics, Probability Theory and Harmonic and Functional Analysis. The seminar is encouraged to be aimed at a broader audience, but it may be of a more specialised nature when indicated in the last line of the abstract. The talks in the seminar are usually 60 minutes, including questions.

At the moment, the seminar takes place via Zoom. See schedule below for links to Zoom meetings.

Should you have any questions or suggestions, feel free to email one of the organizers Jakob Björnberg (jakobbj 'at', Probability Theory), Erik Broman (broman 'at', Probability Theory) or Genkai Zhang (genkai 'at', Analysis).

Coming seminars 

2021-06-15 at 13:15
Augusto Teixera (IMPA, Brazil)

Phase transition for percolation on randomly stretched lattices
In this talk we study the existence/absence of phase transitions for Bernoulli percolation on a class of random planar graphs. More precisely, the graphs we consider have vertex sets given by Z^2 and we start by adding all horizontal edges connecting nearest neighbour vertices. This gives us a disconnected graph, composed of infinitely many copies of Z, with the trivial behavior p_c(Z) = 1. We now add to G vertical lines of edges in {X_i}xZ, where the points X_i are given by an i.i.d. integer-valued renewal process with inter arrivals distributed as T. This graph G now looks like a randomly stretched version of the nearest neighbour graph on Z^2. In this talk we show an interesting phenomenon relating the existence of phase transition for percolation on G with the moments of the variable T. Namely, if E(T^{1+eps}) is finite, then G almost surely features a non-trivial phase transition. While if E(T^{1-eps}) is infinite, then p_c(G) = 1.

This is a joint work with Hilário, Sá and Sanchis.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

Previous seminars

2021-03-23 at 13:15
Nizar Demni (Marseille)

Probabilistic aspects of magnetic Laplacians with uniform fields on certain Riemann surfaces
I shall introduce new discrete probability distributions arising from the spectral resolutions of magnetic Laplacians on the plane and on the hyperbolic disc. These probability distributions are shown to be quasi-infinitely divisible and give rise to new Levy processes. Time permitting I shall also mention the connections of the aforementioned magnetic Laplacians with sub-Riemannian geometry and zeros of random Gaussian series.

2021-03-30 at 13:15
Henna Koivusalo (Bristol)

Sizes of some random limsup and liminf sets
The classical random covering set is the limsup set of balls of random centres. More precisely, given an i.i.d sequence of uniformly distributed centres (\omega_n), and a sequence of radii (r_n) converging to 0, denote by B_n the balls of radius r_n and centres \omega_n. The random covering set E is the set of points that belong to infinitely many of these balls, or \cap_k \cup_{n>k} B_n. This set has been studied for at least a hundred years -- in fact, the Borel-Cantelli lemma was first proven in this context -- and for example, it is known exactly when E has full covering or full measure, and what its dimension is.
We will suggest a far less-studied counterpart to random covering sets: with the centres and radii as above, we study the uniform covering set U, the liminf set of points \cup_{k=1}^n B_{k,n}=\cup_{k=1}^n B(\omega_k, r_n). This set compares to uniformly approximable numbers in Diophantine approximation. We will give conditions on U having full covering, full/null measure, and give bounds on its dimension.
The new results are joint with Lingmin Liao and Tomas Persson.

2021-04-06 at 13:15
Eusebio Gardella (Munster)

The classification problem for free ergodic actions.
One of the basic problems in Ergodic Theory is to determine when two measure-preserving actions of a group on the atomless Borel probability space are orbit equivalent. When the group is amenable, classical results of Dye and Ornstein-Weiss show that any two such actions are orbit equivalent. Thus, the question is relevant only in the non-amenable case. In joint work with Martino Lupini, we showed that for every nonamenable countable discrete group, the relations of conjugacy and orbit equivalence of free ergodic actions are not Borel, thereby answering questions of Kechris. This means that there is in general no method, or uniform procedure, that allows us to determine when two actions of a nonamenable group are conjugate/orbit equivalent. It is a non-classification result, which rules out the existence of any classification theorems which use "nice" (Borel) invariants. The statement about conjugacy also solves the nonamenable case of Halmos' conjugacy problem in Ergodic Theory, originally posed in 1956 for ergodic transformations. The main conceptual innovation is the notion of property (T) for triples of groups, for which a cocycle superrigidity theorem à la Popa can be established. In combination with induction methods developed by Epstein, this is used to obtain a large family of free ergodic actions of the given nonamenable group which have pairwise distinct 1-cohomology groups. No previous knowledge on group amenability will be assumed, and all relevant definitions will be introduced in the course of the presentation.

2021-04-13 at 13:15
Isabelle Tristani (ENS, Paris)

Incompressible Navier-Stokes limit of the Boltzmann equation.
In this talk, we are interested in the link between strong solutions of the Boltzmann and the Navier-Stokes equations. The problem of justifying the connection between mesoscopic and macroscopic equations has been extensively studied. Here, we propose a different approach, intertwining fluid mechanics and kinetic estimates. It enables us to prove convergence of smooth solutions of the Boltzmann equation to solutions to the fluid dynamics equations when the Knudsen number goes to zero. We do not require any smallness at initial time, and our result is valid for any initial data (well prepared or not) in the case of the whole space. We also prove that the time of existence of the solution to the Boltzmann equation is bounded from below by the existence time of the fluid equation as soon as the Knudsen number is small enough. This is a joint work with Isabelle Gallagher.

2021-04-20 at 13:15
Eviatar Procaccia (Technion, Israel)

Dimension of Stationary Hastings Levitov
The Hastings-Levitov process is a planar aggregation process defined by a composition of conformal maps, in which at every time a new particle attaches itself to the existing cluster at a point which is determined by the harmonic measure. The main advantage of this model is that its direct connection to complex analysis makes it tractable. The main disadvantage is non-physical behaviour of the particle sizes. In this talk I will present a new half-plane variant of the Hastings-Levitov model, and will demonstrate that our variant, called the Stationary Hastings-Levitov, maintains the tractability of the original model, while avoiding the non-physical behavior of the particle sizes. Thus this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). Our main result concerns exact growth bounds and fractal dimension.
Based on joint work with Noam Berger and Amanda Turner

2021-05-04 at 13:15
Balazs Rath (Budapest)

A phase transition between endogeny and nonendogeny
The Marked Binary Branching Tree (MBBT) is the family tree of a rate one binary branching process, on which points have been generated according to a rate one Poisson point process, with i.i.d. uniformly distributed activation times assigned to the points. In frozen percolation on the MBBT, initially, all points are closed, but as time progresses points can become either frozen or open. Points become open at their activation times provided they have not become frozen before. Open points connect the parts of the tree below and above it and one says that a point percolates if the tree above it is infinite. We consider a version of frozen percolation on the MBBT in which at times of the form $\theta^n$, all points that percolate are frozen. We show that there exists a $0<\theta^*<1$ such that for $\theta \leq \theta^*$, frozen percolation is almost surely determined by the MBBT but for $\theta > \theta^*$ one needs additional randomness to describe it. Joint work with Jan M. Swart and Márton Szőke.

2021-05-18 at 13:15
Jean Ludwig (Metz)

On the polynomial conjecture for induced representations
Let $G = \exp {{\mathfrak g}}$ be a connected and simply connected real nilpotent Lie group with Lie algebra $\mathfrak g$, $H = \exp {{\mathfrak h}}$ an analytic subgroup of $G$ with Lie algebra $\mathfrak h$, $\chi$ a unitary character of $H$ and $\tau = \text{ind}_H^G \chi$ the unitary representation of $G$ induced by $\chi$. Let $D_{\tau}(G/H)$ be the algebra of the $G$-invariant differential operators on the fiber bundle over the base space $G/H$ associated to the data $(H,\chi)$. We study the polynomial conjecture of Corwin-Greenleaf. Namely, if the monomial representation $\tau$ has finite multiplicities, then the algebra $D_{\tau}(G/H)$ is isomorphic to the algebra ${{\mathbb C}[{\Gamma}_{\tau}]}^H$ of the $H$-invariant polynomial functions on the affine subspace ${\Gamma}_{\tau} = \{\ell \in {\mathfrak g}^*; {\ell}|_{\mathfrak h} = -\sqrt{-1}d\chi\}$ of ${\mathfrak g}^*$. (Joint work with Ali BAKLOUTI, Hidenori FUJIWARA)

2021-05-25 at 13:15
Roger Howe (Yale)

Rank and representations of classical groups over local and finite fields
(Joint Algebraic Geometry/Number Theory and Analysis/Probability Seminar)
The relationship between harmonic analysis on a group and on it subgroups is a natural issue to study in representation theory. In representation theory of reductive algebraic groups, a key example of this is the study of representations induced from parabolic subgroups, which leads to the philosophy of cusp forms, to Harish-Chandra's Plancherel Formula, and to the classification of admissible representations.
Parabolic subgroups are relatively large and have complicated structure. It is worth asking whether fruitful relationships can be found between harmonic analysis on a reductive group and that on relatively simple minded subgroups. A case that has shed considerable light on representations of classical groups is to study the restriction of representations to abelian unipotent radicals. This leads by simple considerations to the idea of rank of representations, which has provided substantial information that is complementary to the approach via parabolic induction. This talk will review results about rank of representations for groups over local fields, and explain how some results can be extended to groups over finite fields.

2021-06-01 at 13:15
Jacob van den Berg (CWI, Amsterdam)

An upper bound on the two-arms exponent for critical percolation on the d-dimensional cubic lattice
We study critical percolation on the d-dimensional cubic lattice. Consider the box of length 2 n centered at 0. The two-arms exponent involves the asymptotic (as n tends to infinity) probability of the event that there are two distinct open clusters connecting neighbours of 0 to the boundary of that box. Such events play a role in several important percolation issues, in particular the uniqueness of the infinite open cluster and the conjectured absence of percolation at the critical point. For d = 2 and for `high' d they have been extensively studied and much is known.
For general d, Cerf pointed out in 2015 that from classical results one can obtain the lower bound 1/2 for the two-arms exponent. By additional interesting arguments he slightly improved that bound.
As far as we know, except for d= 2 and high d no upper bound on the exponent is given in the literature (not even implicitly). We prove the upper bound d^2 + 4 d - 2. Remarkably, part of our proof uses an intermediate result by Cerf which he used for his lower bound.
The talk, which starts with an introduction and then explains the proof, will be quite informal and assumes only a moderate knowledge of percolation theory. Based on joint work with Diederik van Engelenburg.

The seminar will be held via zoom. Members of the department will receive the link and password via mail. Others interested are welcome to attend and can get the link by contacting one of the organizers (please inform us who you are).

2021-06-08 at 13:15
Nina Gantert (Munich)

The TASEP on trees
We study the totally asymmetric simple exclusion process (TASEP) on trees where particles are generated at the root. Particles can only jump away from the root, and they jump from x to y at rate r_{x,y} provided y is empty. Starting from the all empty initial condition, we show that the distribution of the configuration at time t converges to an equilibrium. We study the current and give conditions on the transition rates such that the current is of linear order or such that there is zero current, i.e. the particles block each other. A key step, which is of independent interest, is to bound the first generation at which the particle trajectories of the first n particles decouple. Based on joint work with Nicos Georgiou and Dominik Schmid


Page manager Published: Mon 14 Jun 2021.