TU Budapest -- BME

Next talk scheduled / Következő előadás:

The fractal cylinder model

Abstract

We consider a statistically semi-scale-invariant collection of bi-infinite
cylinders in R^{d}, chosen according to a Poisson line process of intensity
λ. The complement of the union of these cylinders is a random
fractal which we denote by V. This fractal exhibits long-range dependence,
complicating its analysis. Nevertheless, we show that this random fractal
undergoes two different phase transitions. First and foremost we determine
the critical value of λ for which V is non-empty.

We additionally show that for d≧4 this random fractal exhibits a connectivity phase transition in the sense that the random fractal is not totally disconnected for λ small enough but positive.

For d=3 we obtain a partial result showing that the fractal restricted to a fixed plane is always totally disconnected.

An important tool in understanding the connectivity phase transition is the study of a continuum percolation model which we call the fractal random ellipsoid model. This model is obtained as the intersection between the semi-scale-invariant Poisson cylinder model and a k-dimensional linear subspace of R^{d}. Moreover, this model can be understood as a Poisson point
process in its own right with intensity measure Leb_{k} × ξ_{k,d},
where Leb_{k} denotes the Lebesgue measure on R^{k} and
ξ_{k,d} is the shape measure describing the random ellipsoid.

Joint with Erik Broman, Filipe Mussini, Johan Tykesson.

We additionally show that for d≧4 this random fractal exhibits a connectivity phase transition in the sense that the random fractal is not totally disconnected for λ small enough but positive.

For d=3 we obtain a partial result showing that the fractal restricted to a fixed plane is always totally disconnected.

An important tool in understanding the connectivity phase transition is the study of a continuum percolation model which we call the fractal random ellipsoid model. This model is obtained as the intersection between the semi-scale-invariant Poisson cylinder model and a k-dimensional linear subspace of R

Joint with Erik Broman, Filipe Mussini, Johan Tykesson.

2020.02.27 Thursday, 16:15

Further talks planned / További tervezett előadások:

Time Series, Latent Class Analysis, Statistical Modelling and Experimental Design

Abstract

Joint seminar with the Quantitative Social and Management Sciences Research Group

2020.03.02 Monday, 16:15

Unusual place: BME building Q, room QA406.

three lectures of 50 minutes each

local organizers: Bálint Tóth and Gábor Pete. Unusual time!

2020.03.06 Friday, 14:00

Unusual place: Rényi Institute.

TBA

2020.04.09 Thursday, 16:15

Place of the seminar: TU Budapest, building H, room 306. Everybody is welcome!

Az előadások helyszíne: BME H épület 306. Mindenkit szeretettel várunk!

Earlier talks were / Korábbi előadások voltak:

Decoupling inequalities in loop percolation

Abstract

In this talk we are going to talk about a percolation process which arises
from a Poissonian ensemble of simple random walk loops in Z^{d}, for three
dimensions or more. This is a dependent percolation process, states of
vertices have correlations which decay polynomially in the distance between
said vertices. Furthermore, strong decoupling inequalities such as those
available for the gaussian free field or the random interlacements processes
are believed to be false in the loop percolation case. We present a weaker
inequality, which holds for the loop ensemble, and nevertheless allows one
to prove several results for this model which were previously out of reach.

2020.02.20 Thursday, 16:15

Sequential metric dimension for random graphs

Abstract

For a graph G=(V,E), a subset R of V is a resolving set if for every pair
of distinct nodes v_{1}, v_{2} in V there is a (distinguishing) node w in R,
for which the distances d(w,v_{1}) and d(w,v_{2}) are not equal. An equivalent
definition of a resolving set R is that in a two-player localization game,
no matter which node v_{*} Player 1 selects, Player 2 can guess v_{*} only from
the distance queries d(w_{i},v_{*}) for w_{i} in R. The metric dimension (MD) is
defined as the minimum cardinality of R; it can be thought of as the
minimum number of prewritten distance queries Player 2 needs to guess any
v_{*} in V. In this talk we consider the sequential version of the MD (called
SMD), which can be defined using the same localization game, except now the
queries can be adaptive. We are interested how much adaptivity can help to
reduce the number of required queries in G(n,p) random graphs with
p>>log(n)/n. In the first part of the talk, we review the previous results
of Bollobas, Mitsche and Pralat (2012) on the MD of G(n,p), including the
observation that unlike most graph properties, the MD is not monotonic in
p, instead it follows zig-zag behavior. In the second part of the talk, we
introduce the main tools that allow us to extend these results to the SMD,
and quantify reduction of the SMD compared to the MD. We will discuss
connections with source localization of epidemic processes, binary search
on graphs and the birthday paradox.

This is joint work with Patrick Thiran.

This is joint work with Patrick Thiran.

2020.02.13 Thursday, 16:15