TU Budapest -- BME

talks in Spring semester 2015 -- előadásai a 2015 tavaszi félévben

Stationarity, ergodicity and parameter estimation for an affine two-factor process

Abstract

Affine processes are common generalizations of continuous state and
continuous time branching processes with immigration and Ornstein-Uhlenbeck
type processes. Roughly speaking, the affine property means that the
logarithm of the characteristic function of the process at any time is
affine with respect to the initial state. We will consider a special
two-factor affine process given by a jump-type stochastic differential
equation. Based on the asymptotic behavior of the expectation vector, we
introduce a classification of the process by distinguishing subcritical,
critical and supercritical cases. In the subcritical case we prove that the
affine model in question has a unique strictly stationary solution, further,
in the subcritical diffusion case, ergodicity is also shown. As an
application, in the latter case, we study asymptotic behaviour of the
maximum likelihood estimator of the drift parameters of the process based on
continuous time observations proving strong consistency and asymptotic
normality.

Joint work with Leif Doring, Zenghu Li and Gyula Pap.

Joint work with Leif Doring, Zenghu Li and Gyula Pap.

2016.06.16 Thursday, 16:15

Speeding up non-Markovian First Passage Percolation by a single extra edge

Abstract

One model of real-life spreading processes is First Passage Percolation
(also called SI model) on random graphs. Social interactions often follow
bursty patterns, which are usually modelled with non-Markovian heavy-tailed
edge weights. On the other hand, random graphs are often locally tree-like,
and spreading on trees is very slow, because of bottleneck edges with huge
weights. We show the surprising phenomenon that adding a single random edge
to a tree typically accelerates the process severely. We examine this
acceleration effect on some natural models of random trees: critical
Galton-Watson trees conditioned to be large in some way, uniform random
trees using Loop Erased Random Walks and Pólya urn ideas, and will also
discuss what should happen on near-critical Erdős-Rényi graphs.

This is joint work with Gábor Pete.

This is joint work with Gábor Pete.

2016.06.09 Thursday, 16:15

Kockavázak uniójának Lebesgue mértéke és Hausdorff dimenziója / Lebesgue measure and Hausdorff dimension of skeletons of cubes

Abstract

Legyen B olyan halmaz R^3-ben (vagy általánosabban R^n-ben), amely a tér
minden pontja körül (mint középpont körül) tartalmaz kocka élvázat (vagy
általánosabban n-dimenziós kocka k-dimenziós vázát). Három esetet
vizsgálunk:

(1) csak tengelypárhuzamos kockát engedünk meg,

(2) bármilyen kockát megengedünk, azaz elforgatottat is,

(3) csak egységkockákat engedünk meg, vagyis csak forgatni tudunk.

Az fog kiderülni, hogy B mindhárom esetben lehet nullmértékű, sőt a legtöbb esetben a Hausdorff dimenzió lehet n-nél kevesebb. A konstrukciók a Baire kategória tételt használják: azt mutatjuk meg, hogy tipikus jó elrendezés esetén B kicsi.

Alan Chang-gel, Csörnyei Mariannával és Héra Kornéliával közös eredmények.

(1) csak tengelypárhuzamos kockát engedünk meg,

(2) bármilyen kockát megengedünk, azaz elforgatottat is,

(3) csak egységkockákat engedünk meg, vagyis csak forgatni tudunk.

Az fog kiderülni, hogy B mindhárom esetben lehet nullmértékű, sőt a legtöbb esetben a Hausdorff dimenzió lehet n-nél kevesebb. A konstrukciók a Baire kategória tételt használják: azt mutatjuk meg, hogy tipikus jó elrendezés esetén B kicsi.

Alan Chang-gel, Csörnyei Mariannával és Héra Kornéliával közös eredmények.

2016.06.02 Thursday, 16:15

Large-scale invariance in percolation models (with strong correlations)

Abstract

I will discuss recent progress in understanding
supercritical percolation models on lattices, particularly in the
presence of strong spatial correlations. This includes quenched
Gaussian heat kernel bounds, Harnack inequalities, and local CLT for
the random walk on infinite percolation clusters. The results apply to
the random interlacements at all levels, the vacant set of random
interlacements and the level sets of the Gaussian free field in the
regime of local uniqueness.

2016.05.12 Thursday, 16:15

Small world phenomena for Random Apollonian Networks

Abstract

Random Apollonian networks are graphs which can be constructed as
a sort of dual to the Apollonian circle packing problem. They have a
tractable hierarchical structure. We show how to exploit this to prove a
central limit theorem for the shortest path between two randomly chosen
vertices of the graph.

Joint work with Júlia Komjáthy and Lajos Vágó.

Joint work with Júlia Komjáthy and Lajos Vágó.

2016.04.28 Thursday, 16:15

Correlation bounds and one-ended tail triviality for factors of IID on trees

Abstract

We study factor of i.i.d. processes on the d-regular tree. We show
that if such a process is restricted to two distant connected subgraphs of
the tree, then the two parts are basically uncorrelated. More precisely, we
give an explicit bound for the correlation of any measurable functions of
the two parts, the bound only depending on d and the distance of the
subgraphs. This result can be considered as a quantitative version of the
fact that factor of i.i.d. processes have trivial 1-ended tails.

Joint work with Ágnes Backhausz, Balázs Gerencsér, and Máté Vizer.

Joint work with Ágnes Backhausz, Balázs Gerencsér, and Máté Vizer.

2016.04.21 Thursday, 16:15

Application of the Furstenberg measure for self-affine systems

Abstract

The Furstenberg measure is the stationary measure of a
Markov chain on the projective space. The geometric measure
theoretical properties of the Furstenberg measure allows us to give
sufficient conditions for calculating the dimension of self-affine
sets and measures, in a very natural way. We will see how.

2016.04.14 Thursday, 16:15

Explosive branching processes and their applications to epidemics and distances in power-law random graphs

Abstract

In this talk I will give investigate the question of
explosion of branching processes, i.e., when is it possible that a BP
produces infinitely many offspring in finite time. Two important
cases in terms of application are age-dependent BPs and BPs arising
from epidemic models where individuals are only contagious in a
possibly random interval after being infected. This imposes
dependencies between the birth-time of the children of an individual.

The motivation for studying the explosiveness question is to understand weighted distances in locally tree-like random graphs, such as the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and thus the initial stages of the spreading can be the approximated by an infinite mean offspring BP. I will explain the recent results on this area. This part is joint work with Enrico Baroni and Remco van der Hofstad.

The motivation for studying the explosiveness question is to understand weighted distances in locally tree-like random graphs, such as the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and thus the initial stages of the spreading can be the approximated by an infinite mean offspring BP. I will explain the recent results on this area. This part is joint work with Enrico Baroni and Remco van der Hofstad.

2016.04.07 Thursday, 16:15

Non-intersecting squared Bessel paths at the hard edge tacnode / Egymást nem metsző négyzetes Bessel-folyamatok az érintési pont közelében

Abstract

We consider a system of non-intersecting squared Bessel processes
which all start from one point at time 0 and they all return to the
same point at time 1. Under the scaling of the starting and ending
points when the macroscopic boundary of the paths touches the hard
edge, we describe the limiting critical process in the neighbourhood
of the touching point. In the talk, I will introduce the notion of
non-intersecting paths and determinantal processes showing the
connection to random matrices.

Joint work with Steven Delvaux.

Joint work with Steven Delvaux.

2016.03.10 Thursday, 16:15

A conformally invariant growth process of SLE excursions / Egy SLE-kirándulásokból építkező konform-invariáns növekedési folyamat

Abstract

Motivated by Diffusion Limited Aggregation being too hard to
understand, Itai Benjamini suggested a way to construct a fully
conformally invariant growth process in the plane: an aggregation
process of chordal SLE(κ) excursions in the unit disk, starting
from the boundary, growing towards all inner points simultaneously,
invariant under all conformal self-maps of the disk. We prove that
this conformal growth process of excursions, abbreviated as
CGE(κ), exists iff κ∈ [0,4), and that, maybe sadly, it
does not create additional fractalness: the Hausdorff dimension of the
closure of all the SLE(κ) arcs attached is 1+κ/8 almost
surely.

I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).

Joint work with Hao Wu.

I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).

Joint work with Hao Wu.

continued from last week / a múlt heti előadás folytatása

2016.03.03 Thursday, 16:15

A conformally invariant growth process of SLE excursions / Egy SLE-kirándulásokból építkező konform-invariáns növekedési folyamat

Abstract

Motivated by Diffusion Limited Aggregation being too hard to
understand, Itai Benjamini suggested a way to construct a fully
conformally invariant growth process in the plane: an aggregation
process of chordal SLE(κ) excursions in the unit disk, starting
from the boundary, growing towards all inner points simultaneously,
invariant under all conformal self-maps of the disk. We prove that
this conformal growth process of excursions, abbreviated as
CGE(κ), exists iff κ∈ [0,4), and that, maybe sadly, it
does not create additional fractalness: the Hausdorff dimension of the
closure of all the SLE(κ) arcs attached is 1+κ/8 almost
surely.

I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).

Joint work with Hao Wu.

I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).

Joint work with Hao Wu.

2016.02.25 Thursday, 16:15

Some problems related to the additive representation functions

PhD defense / PhD disszertáció nyilvános vitája

2016.02.03 Wednesday, 10:30

Unusual place: BME building H, room 607.