### Department of Mathematics and Systems Analysis

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Aalto Stochastics and Statistics Seminar is organized by Kalle Kytölä, Lasse Leskelä, and Pauliina Ilmonen. Feel free to contact one of us if you are interested in giving a talk. You may also earn credit points by active participation.

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- 12.8. 11:00 Dr Vinay Kumar BR (INRIA Sophia Antipolis): A probabilistic broadcast mechanism on random geometric graphs – M3 (M234)
We consider the problem of energy-efficient broadcasting on homogeneous random geometric graphs (RGGs) within a large finite box around the origin. A source node at the origin encodes $k$ data packets of information into $n\ (>k)$ coded packets and transmits them to all its one-hop neighbors. The encoding is such that, any node that receives at least $k$ out of the $n$ coded packets can retrieve the original $k$ data packets. Every other node in the network follows a probabilistic forwarding protocol; upon reception of a previously unreceived packet, the node forwards it with probability $p$ and does nothing with probability $1-p$. We are interested in the minimum forwarding probability which ensures that a large fraction of nodes can decode the information from the source. We deem this a \emph{near-broadcast}. The performance metric of interest is the expected total number of transmissions at this minimum forwarding probability, where the expectation is over both the forwarding protocol as well as the realization of the RGG. In comparison to probabilistic forwarding with no coding, our treatment of the problem indicates that, with a judicious choice of $n$, it is possible to reduce the expected total number of transmissions while ensuring a near-broadcast. Techniques from continuum percolation and ergodic theory are used to characterize the probabilistic broadcast algorithm. Joint work with Navin Kashyap and D. Yogeshwaran

- 30.5. 14:15 Elmer Bergman (Aalto University): Connectivity of passive random intersection graphs and their intersection with ErdősRényi graphs – M3 (M234)
This thesis studies the connectivity of passive random intersection graphs. In addition to this, it studies the connectivity of an intersection between a passive random intersection graph and an ErdősRényi graph. Random intersection graphs can be used to model many real-life phenomena. For example, social networks and communication in sensor networks can be modelled by random intersection graphs. A random intersection graph is a random graph, where nodes are assigned attributes according to some random process. Two nodes are connected by an edge if they have at least one attribute in common. For a passive random intersection graph, each attribute is given a number according to some probability distribution. Each attribute then chooses that number of nodes, uniformly at random from the whole set of nodes. The chosen nodes are given the respective attribute. Two nodes are thus connected, if at least one attribute chooses them both. This thesis presents zero-one laws on passive random intersection graphs being connected and not having isolated nodes. This thesis also presents zero-one laws on the intersection between a passive random intersection graph and an ErdősRényi graph, being connected and not having isolated nodes.

- 14.1. 14:00 Olle Hallqvist Elias (Alfréd Rényi Institute of Mathematics): Interlacements and the GFF – Zoom
- 14.1. 11:00 Ommolbanin Behzad (University of Isfahan): Exterior powers, Polynomial rings, and Representation of Lie Algebras – Zoom
I will report on some recent work of myself, A.Contiero and D. Martins about representing lie algebras of vector space endomorphisms on exterior algebras, seeing it as the finite type case of the celebrated DJKM bosonic vertex operator representation of gl_∞(Q).

- 4.1. 17:00 Olga Chekeres (University of Connecticut): Quantum Wilson surfaces and topological interactions – Zoom
- 3.1. 11:00 Julien Roussillon (KTH): Confluence of correlation functions in Liouville theory – Zoom
- 26.11.2021 10:15 Augustin Lafay (ENS): Web models as generalizations of statistical loop models – Zoom
Two dimensional gases of non intersecting loops have been a subject of study in mathematical physics for more than thirty years because of their numerous connections to integrability, two dimensional conformal field theory, random geometry and combinatorics. In this talk, I will present a natural generalization of loop models to gases of graphs possessing branchings. These graphs are called webs and first appeared in the mathematical community as diagrammatic presentations of categories of representations of quantum groups. The web models posses properties similar to the loop models. For instance, it will be shown that they describe, for some tuning of the parameters, interfaces of spin clusters in Zn spin models. Focusing on the numerically more accesible case of Uq(sl3) webs (or Kuperberg webs), it is possible to identify critical phases that are analogous to the dense and dilute phases of the loop models. These phases are then described by a Coulomb Gas with a two component bosonic field.

- 15.11.2021 14:15 Yizheng Yuan: Refined regularity of SLE – Y229a
SLE (Schramm-Loewner evolution) is a family of random planar curves that have some natural conformal invariance properties. They appear in a variety of planar models that exhibit conformal invariance in the scaling limit. Regarding their regularity, the optimal Hoelder and p-variation exponents are known from previous works. In this talk, I will present refinements of the regularity statements to the logarithmic scale. I will present a new argument for obtaining these results and discuss some applications.

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