Self-avoiding walk of length n on the integer lattice Z^d is the uniform measure on nearest-neighbour walks in Z^d that begin at the origin and are of length n. If such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. The result of doing so is a self-avoiding polygon. We investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. Developing a method (the "snake method") employed in joint work with Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most n^{-1/2 + o(1)} in any dimension at least two. Developing a quite different method of polygon joining employed by Madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. We further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability of n^{-4/7 + o(1)} on a subsequence in the two-dimensional setting.

We review and put into context the mixing properties of standard Markov chains and variants for Small World Networks based on the model of Newman and Watts.
Consensus problems provide a natural guideline to optimize Markov chains with uniform stationary distribution. We learn what is achievable for reversible Markov chains when the underlying graph is a Small World Network.
Without the reversibility condition comes additional flexibility but also complexity. We show an example where mixing turns out to be substantially faster, confirming that it is an area worth addition investigation.
Furthermore, the cutoff phenomenon of mixing for this setup is demonstrated.

A pénzügyi rendszerek stressz szintjének nyomon követése fontos a Makroprudenciális Politika és pénzügyi stabilitás szempontjából. Különösen igaz ez a globális pénzügyi válság után. A pénzügyi stressz indexek nagy frekvenciájú hangulati indikátorok, melyek a pénzügyi rendszerben fennálló feszültséget mérik, lehetővé téve a szabályozó hatóság számára, hogy a megfelelő lépéseket jó időben megtéve minimalizálja a válság reálgazdasági következményeit. A prezentáció célja, hogy bemutassa a Magyar Nemzeti Bank ez irányú erőfeszítéseit. Az itt bemutatott pénzügyi stressz index faktor modell segítségével a rendelkezésre álló adatokat próbálja informatív és naprakész módon tömöríteni.

A munka a bankközi fertőzések hatásait vizsgálja bankok hitelezési hálózatain. Bemutatunk egy új mérőszámot Acemoglu et al. (2015) alapján, majd az ő eredményeik által motiválva, a mérőszámot olyan jól ismert centralitási mutatókhoz hasonlítjuk, amelyeket már használnak a szakirodalomban, de nem veszik figyelembe a fertőzési mechanizmus struktúráját. Meghatározzuk a mérettel korrigált harmonikus távolságok explicit formuláját és kiterjesztjük a fogalmat heterogén bankközi rendszerre. A numerikus eredmények szerint ez a mutatószám nem teljesít jobban, mint a korábban alkalmazott egyszerűbb centralitások, de alkalmas a fontos intézmények azonosítására, időbeli alakulása pedig jelzi a bankközi piacon a stressz szituációt.

Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of phase transitions.

Joint work with Jean-Christophe Mourrat (ENS Lyon).

It is known, since the work of Dynkin building on pioneering ideas of Symanzik, that Markov processes and Markovian random fields are deeply related. In the simplest discrete setup of graphs, on which we shall focus in this talk, this relation links random walk paths to the discrete Gaussian free field.

At the formal level, this relation stems from the fact that a same operator, the discrete Laplacian, generates two types of processes: (1) by means of its semi-group (this is the random walk), (2) by means of its Dirichlet form (this is the discrete Gaussian free field).

At a refined probabilistic level, it is known that the local time (i.e. the time spent at each vertex) of random walk paths is related to the square of the discrete Gaussian free field. This is Dynkin's isomorphism, which admits further variations and extensions, including theorems of Eisenbaum, Le Jan, and Sznitman.

In this talk, I will consider more general functionals of a path than its local time. These encode more faithfully the actual geometry of the trajectory and I will explain how they are related to vectorial versions of the Gaussian free field. Time permitting, I will say a few words about the related topics of gauge theory. No prior knowledge will be assumed. Joint work with Thierry Lévy (Univ. Paris 6).

When two scatterers intersect tangentially in a planar dispersing billiard domain, the particle can be trapped in the cuspoidal region. These phenomena are known to cause slow decay of correlations and non-standard limit theorems. In joint work with N. Chernov and D. Dolgopyat we have shown that the second moments of the appropriately normalized Birkhoff sums converge to a value which is twice the second moment of the limit distribution. In my talk I would like to describe this doubling effect and show that it arises in many other situations, in particular in an entirely probabilistic setting.

The goal of the talk is to show how the graph limit approach can be used to understand spectral properties of large random regular graphs. A large random regular graph has large girth with high probability, and locally looks like a tree. To put it in another way, a sequence of random regular graphs converges to the infinite regular tree in the Benjamini--Schramm (local) sense of graph convergence. In our work, we consider invariant random processes on the infinite tree that satisfy the eigenvalue equation. As a consequence of the results in the infinite setting, we could prove that delocalized eigenvectors of finite random regular graphs are close to Gaussian.

Joint work with Balázs Szegedy.