TU Budapest -- BME

talks in Spring semester 2019 -- előadásai a 2019 tavaszi félévben

Speed of convergence of the average of certain random matrices

Abstract

John's theorem states that if the Euclidean unit ball is the largest volume
ellipsoid in a convex body K in R^{d}, then there is a set of unit
vectors u_{1},...,u_{m} on the boundary of K such that the identity
operator I on R^{d} is a positive linear combination of the diads
u_{i}⊗u_{i}. Put in another way, I is the expectation of a probability
distribution on the set of n×n real matrices supported on a certain
set of rank one matrices.

Motivated by geometric applications, it is natural to ask if the average of few of these random matrices is close to I. Our main interest is whether the known positive answer to this question extends from diads to larger classes of matrices.

Joint work with Grigoriy Ivanov and Alexander Polyanskii.

Motivated by geometric applications, it is natural to ask if the average of few of these random matrices is close to I. Our main interest is whether the known positive answer to this question extends from diads to larger classes of matrices.

Joint work with Grigoriy Ivanov and Alexander Polyanskii.

2019.06.13 Thursday, 16:15

Fractals in dimension theory and complex networks

PhD public defense

szokatlan időpont / unusual time

szokatlan időpont / unusual time

2019.06.05 Wednesday, 16:15

Unusual place: room H607.

Level-set percolation of the Gaussian free field on large d-regular expanders

Abstract

In this joint project with Jiří Černý we study level-set percolation of the
zero-average Gaussian free field on a class of large d-regular graphs with d
larger equal 3, containing d-regular expanders of large girth and typical
realisations of random d-regular graphs. Through suitable local
approximations of the zero-average Gaussian free field by the Gaussian free
field on the infinite d-regular tree we are able to establish a phase
transition for level-set percolation of the zero-average Gaussian free field
which occurs at the critical value for level-set percolation in the infinite
model.

unusual time / szokatlan időpont

2019.06.03 Monday, 16:15

Unusual place: Rényi Intézet, tondós terem.

The mathematics of asymptotic stability in the Kuramoto model

Abstract

The Kuramoto model is the archetype of nonlinear heterogeneous systems of
coupled oscillators. Its phenomenology (in the continuum limit) strongly
relies on the nonlinear stability of its stationary states. To understand
and to rigorously assert stability in this infinite-dimensional setting have
been long-standing challenges, and show similar features of the Landau
damping in the Vlasov equation. In this talk, I will review results on
stability conditions and asymptotic stability of various stationary states,
that mathematically confirm the intuited phenomenology and its dependence on
parameters.

unusual time / szokatlan időpont; joint with the Dynamical Systems seminar

2019.05.30 Thursday, 17:15

Asymptotic properties of mean field coupled maps

PhD public defense

szokatlan időpont / unusual time

szokatlan időpont / unusual time

2019.05.28 Tuesday, 14:00

Unusual place: room H607.

Inhomogeneous percolation on ladder graph

Abstract

We define an inhomogeneous percolation model on `ladder graphs' obtained as
direct products of an arbitrary graph G=(V,E) and the set of integers
(vertices are thought of as having a `vertical' component indexed by an
integer). We make two natural choices for the set of edges, producing an
non-oriented and an oriented graph. These graphs are endowed with
percolation configurations in which independently, edges inside a fixed
infinite `column' are open with probability q, and all other edges are open
with probability p. For all fixed q one can define the critical percolation
threshold p_{c}(q). We show that this function is continuous in (0,1). Joint
work with D. Valesin.

2019.05.09 Thursday, 16:15

Asymptotic behavior of random walks and growth of groups

Abstract

The question about existence of groups of intermediate growth
(super-polynomial but sub-exponential) was raised by Milnor in the 60s.
First examples of such groups were constructed by Grigorchuk in the early
1980s. We will discuss some probabilistic ideas in studying such groups, and
explain near optimal volume growth lower estimates of Grigorchuk groups
coming from random walks with nontrivial Poisson-Furstenberg boundary on
these groups. Joint with Anna Erschler.

2019.05.02 Thursday, 16:15

Computation of the critical point for the random-cluster model on Z

Abstract

The random-cluster model (or Fortuin-Kasteleyn percolation) plays a key role
in studies of models on lattices, as it is connected to many of them, and
the results obtained for RCM can be though applied for other models.

In this talk I will present another proof of the well-known fact that for the square lattice the critical probability of the random-cluster model p_{cr} is equal to √q/(1+√q) for q in [1,4]. Unlike other
proofs, this one involves the method of parafermionic observables applied to
exploration paths in boxes and strips of growing size.

This result was presented in a joint work with E. Mukoseeva during my PhD under the supervision of H. Duminil-Copin.

In this talk I will present another proof of the well-known fact that for the square lattice the critical probability of the random-cluster model p

This result was presented in a joint work with E. Mukoseeva during my PhD under the supervision of H. Duminil-Copin.

2019.04.25 Thursday, 16:15

Algorithmic Pirogov-Sinai Theory

Abstract

What does a random independent set look like? This is an important problem
at the intersection of probability theory, statistical mechanics, and
theoretical computer science. I will introduce this problem, also known as
the hard-core model, and explain various ways in which the question can be
answered. In particular, I will describe a recent algorithm for producing
approximate samples of high-density independent sets on lattices. This is
the first known algorithm in the high-density regime, and standard
algorithms, like MCMC using Glauber dynamics, are known to fail.

Based on joint work with Will Perkins and Guus Regts.

Based on joint work with Will Perkins and Guus Regts.

2019.04.11 Thursday, 16:15

20 éves a BME Sztochasztika Szeminárium

This
seminar event overlapped (in time) with a demonstartion in support of
the independence of Hungarian academic research institutions. This
overlap was by chance and unintended: the seminar was announced much
earlier. We are sorry for the clash.

2019.03.21 Thursday, 16:15

Unusual place: room H406.

Információs vetületek geometriája és általánosított ML becslések

2019.03.21 Thursday, 16:25

Unusual place: room H406.

Asymptotic properties of mean field coupled maps - PhD home defense / PhD házivédés

Abstract

We study mean field coupled map systems of uniformly expanding circle
maps. We first consider N globally coupled doubling maps of the circle
with diffusive coupling. Reconsidering and extending the results of
Fernandez we prove ergodicity breaking for N=3 and N=4 and showcase
some synchronization phenomena for various values of N in case of
strong coupling. We then introduce the continuum limit of the system,
where we generalize the doubling map to a smooth uniformly expanding
circle map T. Now the state of the system is described by a density
function and the evolution of an initial density with respect to the
transfer operator of the coupled dynamics is studied. We show that for
weak enough coupling, a unique, asymptotically stable invariant
density exists in a suitable function space. Furthermore, we show that
this invariant density depends Lipschitz continuously on the coupling
parameter. For sufficiently strong coupling, we prove convergence to
a point mass which can be interpreted as chaotic synchronization. To
conclude, we provide some outlook on the case of discontinuous T.

unusual time / szokatlan időpont

2019.03.08 Friday, 14:15

Level set percolation for Gaussian fields

Abstract

In this talk, we consider a random smooth Gaussian function from the
plane to ℝ and, given a level u, we colour the points where the
function is larger than u in black and the points where the function
is less than u in white. By relying on recent works by V. Beffara and
D. Gayet, we study the percolation properties of this random colouring
and try to compare it with Bernoulli percolation. Joint works with S.
Muirhead and A. Rivera.

2019.03.07 Thursday, 16:15

Dynamical Voronoi percolation

Abstract

Consider a Poisson point process in the plane, construct its Voronoi
tiling, and colour each tile in black with probability 1/2 and in
white with probability 1/2. This defines a critical Voronoi
percolation configuration. We prove that, if we let each point move
according to a long range stable Lévy process, then there exist
atypical (random) times with an unbounded monochromatic component. To
this purpose, we study a continuous spectral object - the annealed
spectral sample - which is a continuous analogue of the spectral
sample studied by Garban, Pete and Schramm.

unusual time / szokatlan időpont - this is a seminar of the Rényi institute

2019.03.04 Monday, 17:15

Unusual place: Rényi Institute, angyalkás terem.

PhD home defense: Fractals in dimension theory and complex networks

Abstract

The main aim of the Thesis is to demonstrate the diverse applicability
of fractals in different areas of mathematics. Namely,

Thesis advisor: Károly Simon

- widen the class of planar self-affine carpets for which we can calculate the different dimensions especially in the presence of overlapping cylinders,
- perform multifractal analysis for the pointwise Hölder exponent of a
family of continuous parameterized fractal curves in R
^{d}including deRham's curve, - show how hierarchical structure can be used to determine the asymptotic growth of the distance between two vertices and the diameter of a random graph model, which can be derived from the Apollonian circle packing problem.

Thesis advisor: Károly Simon

unusual time / szokatlan időpont

2019.02.01 Friday, 13:15

Unusual place: room H406.

The Directed Landscape

Abstract

The longest increasing subsequence in a random permutation, the second
class particle in TASEP, and semi-discrete polymers at zero temperature
have the same scaling limit: a random function with Holder exponent
2/3. This limit can be described in terms of the directed landscape, a
random metric at the heart of the Kardar-Parisi-Zhang universality class.

In this talk I will give some insight into the construction of the directed landscape, which is joint work with Duncan Dauvergne and Janosch Ortmann.

In this talk I will give some insight into the construction of the directed landscape, which is joint work with Duncan Dauvergne and Janosch Ortmann.

2019.01.31 Thursday, 16:15