TU Budapest -- BME

talks in Spring semester 2018 -- előadásai a 2018 tavaszi félévben

Turning a coin over instead of tossing it

Abstract

Given a sequence of numbers p_{n} in [0,1], consider the
following experiment. First, we flip a fair coin and then, at step n,
we turn the coin over to the other side with probability p_{n}, n>1,
independently of the sequence of the previous terms. What can we say
about the distribution of the empirical frequency of heads as
n→∞?
We show that a number of phase transitions take place as the turning
gets slower (i.e. p_{n} is getting smaller), leading first to the
breakdown of the Central Limit Theorem and then to that of the Law of
Large Numbers. It turns out that the critical regime is p_{n}=const/n.
Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and
Arcsine laws.
The critical regime is particularly interesting: when the
corresponding random walk is considered, an interesting process
emerges as the scaling limit; also, a connection with Polya urns will
be mentioned.
This is joint work with S. Volkov (Lund) and Z. Wang (Boulder). See
also the paper
https://drive.google.com/file/d/0B4ZkCm_J6qB8NXNRcUFPM0hqRUU/view

2018.06.21 Thursday, 16:15

High-density hard-core model on a triangular lattice

Abstract

The hard-core model has attracted attention for quite a long time; the
first rigorous results about the phase transition on a lattice were
obtained by Dobrushin in late 1960s. Since then, various aspects of
the model gained importance in a number of applications. We propose a
solution for the high-density hard-core model on a triangular lattice.
The high-density phase diagram (i.e., the collection of pure phases)
depends on arithmetic properties of the exclusion distance D; a
convenient classification of possible cases can be given in terms of
Eisenstein primes. For two classes of values of D the phase diagram
is completely described: (I) when either
D or D/√3 is a
positive integer whose prime decomposition does not contain factors of
the form 6k+1, (II) when D^{2} is an integer whose prime
decomposition contains (i) a single prime of the form 6k+1, and (ii)
other primes, if any, in even powers, except for the prime 3. For
the remaining values of D we offer some partial results. The main
method of proof is the Pirogov-Sinai theory with an addition of
Zahradnik's argument. The theory of dominant ground states is also
extensively used, complemented by a computer-assisted argument.

This is a joint work with A. Mazel and Y. Suhov.

This is a joint work with A. Mazel and Y. Suhov.

unusual time / szokatlan időpont

2018.06.15 Friday, 14:15

Random Recursive Tree, Branching Markov Chains and Urn Models

Abstract

In this talk, we will establish a connection between random recursive tree,
branching Markov chain and urn model. Exploring the connection further we
will derive fairly general scaling limits for urn models with colors indexed
by a Polish Space and show that several exiting results on
classical/non-classical urn schemes can be easily derived out of such
general asymptotic. We will further show that the connection can be used to
derive exact asymptotic for the sizes of the connected components of a
random recursive forest, obtained by removing the root of a random recursive
tree.

This is a joint work with Debleena Thacker.

This is a joint work with Debleena Thacker.

2018.06.14 Thursday, 16:15

On the fundamental understanding of divide-and-conquer methods (Az 'oszd meg és uralkodj' algoritmusok általános vizsgálata)

Abstract

In recent years, the amount of available information has become so
vast in certain fields of applications that it is infeasible or
undesirable to carry out the computations on a single server. This has
motivated the design and study of distributed statistical or learning
methods. In distributed methods, the data is split amongst different
administrative units and computations are carried out locally in
parallel to each other. The outcome of the local computations are then
aggregated into a final result on a central machine.

First, we will compare the theoretical properties of various (Bayesian) distributed methods proposed in the literature on the benchmark signal in Gaussian white noise model. Then we consider the limitations and guarantees of distributed methods in general under communication constraints on the same benchmark nonparametric model.

This is an ongoing joint work with Harry van Zanten.

First, we will compare the theoretical properties of various (Bayesian) distributed methods proposed in the literature on the benchmark signal in Gaussian white noise model. Then we consider the limitations and guarantees of distributed methods in general under communication constraints on the same benchmark nonparametric model.

This is an ongoing joint work with Harry van Zanten.

2018.06.07 Thursday, 16:15

The continuous-time lace expansion and its applications

Abstract

The lace expansion is one of the primary tools for proving that
probability models in high dimensions have mean field behaviour. I
will explain the previous sentence by describing joint work in
progress with David Brydges and Mark Holmes in which we develop a
continuous time lace expansion. To motivate our methods I will
introduce a class of n-component field theories that are
generalizations of the Ising model of ferromagnetism. When n is zero,
one, or two we are able to analyze these models. Our results for the
case n=2 are new.

2018.05.31 Thursday, 16:15

Scaling exponents for high-dimensional spanning forests and sandpiles

Abstract

The uniform spanning forests (USFs) of infinite an infinite graph
G are defined as infinite volume limits of uniform spanning trees on finite
subgraphs of G. In this talk, I will describe how we use a new way of
sampling the USF using the random interlacement process to compute various
critical exponents governing the large-scale geometry of trees in the forest
in a wide variety of â€śhigh-dimensionalâ€ť graphs, including Z^{d} for d <= 5
and every bounded degree nonamenable graph. I will then sketch how this
allows us to compute related exponents describing the geometry of avalanches
in the Abelian sandpile model on the same class of graphs.

2018.05.17 Thursday, 16:15

Recurrence of the vertex-reinforced jump process in two dimensions

Abstract

The vertex-reinforced jump process (VRJP) is a linearly reinforced random
walk in continuous time. Reinforcing means that the VRJP prefers to jump to
vertices it has previously visited; the strength of the reinforcement is a
parameter of the model.

Sabot and Tarres have shown that the VRJP is related to a model known as the H22 supersymmetric hyperbolic spin model, which originated in the study of random band matrices. By making use of results for the H22 model they proved the VRJP is recurrent for sufficiently strong reinforcement. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for all reinforcement strengths.

This talk is about joint work with R. Bauerschmidt and A. Swann.

Sabot and Tarres have shown that the VRJP is related to a model known as the H22 supersymmetric hyperbolic spin model, which originated in the study of random band matrices. By making use of results for the H22 model they proved the VRJP is recurrent for sufficiently strong reinforcement. I will present a new and direct connection between the VRJP and hyperbolic spin models (both supersymmetric and classical), and show how this connection can be used to prove that the VRJP is recurrent in two dimensions for all reinforcement strengths.

This talk is about joint work with R. Bauerschmidt and A. Swann.

2018.03.29 Thursday, 16:15

Conditional measure on the Brownian path and other random sets - Part 2

Abstract

Let B denote the range of the Brownian motion in R^{d}. For a deterministic
Borel measure ν we wish to find a random measure μ such that the support
of μ is contained in B and the expectation of μ is ν. We discuss when
exactly we can find such a μ.

unusual time / szokatlan időpont

2018.03.23 Friday, 14:15

Establishing and maintaining databases of self-affine tiles

Abstract

unusual time / szokatlan időpont; sharp start

2018.03.23 Friday, 13:00

Unusual place: BME H épület 406.

Conditional measure on the Brownian path and other random sets - Part 1

Abstract

Let B denote the range of the Brownian motion in R^{d}. For a deterministic
Borel measure ν we wish to find a random measure μ such that the support
of μ is contained in B and the expectation of μ is ν. We discuss when
exactly we can find such a μ.

2018.03.22 Thursday, 16:15

Multiple interaction strategies, parameter estimation, and clustering in networks

PhD public defense

szokatlan időpont / unusual time

szokatlan időpont / unusual time

2018.03.09 Friday, 14:00

Criticality in random transposition random walk

Abstract

The random walk on the permutations of [N] generated by the transpositions
was shown by Diaconis and Shahshahani to mix with sharp cutoff around N log
N /2 steps. However, Schramm showed that the distribution of the sizes of
the largest cycles concentrates (after rescaling) on the Poisson-Dirichlet
distribution PD(0,1) considerably earlier, after (1+ε)N/2 steps. We
show that this behaviour truly emerges precisely during the critical window
ofÂ (1+λ N^{-1/3}) N/2 steps, as λ →∞. Our
methods are rather different, and involve an analogy with the classical
Erdos-Renyi random graph process, the metric scaling limits of a
uniformly-chosen connected graph with a fixed finite number of surplus
edges, and analysing the directed cycle structure of large 3-regular graphs.

Joint work with Christina Goldschmidt.

Joint work with Christina Goldschmidt.

2018.03.08 Thursday, 16:15

Invariance principle for the random Lorentz gas beyond the Boltzmann-Grad (or, Gallavotti-Spohn) limit - Part 2

Abstract

Let hard ball scatterers of radius r be placed in R^{d}, centred
at the points of a Poisson point process of intensity ρ. The volume
fraction r^{d}ρ is assumed to be sufficiently low so that with positive
probability the origin is not trapped in a finite domain fully surrounded by
scatterers. The Lorentz process is the trajectory of a point-like particle
starting from the origin with randomly oriented unit velocity subject to
elastic collisions with the fixed (infinite mass) scatterers. The question
of diffusive scaling limit of this process is one of the major open problems
in classical statistical physics.

Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r^{d-1}ρ→1 and the time scale is fixed, the Lorentz process (described
informally above) converges to a Markovian random flight process, with
independent exponentially distributed free flight times and Markovian
scatterings. It is essentially straightforward to see that taking a second
diffusive scaling limit (after the Gallavotti-Spohn limit) yields invariance
principle.

I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r^{d-1}ρ→1 (as in B-G) and simultaneously rescaling time by
T∼r^{-2+ε} we prove invariance principle (under diffusive
scaling) for the Lorentz trajectory. (Note that the B-G limit and diffusive
scaling are done simultaneously and not in sequel.) The proof is essentially
based on control of the effect of recollisions by probabilistic coupling
arguments. The main arguments are valid in d=3 but not in d=2.

Joint work (partly in progress) with Chris Lutsko (Bristol).

Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r

I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r

Joint work (partly in progress) with Chris Lutsko (Bristol).

unusual time / szokatlan időpont

2018.02.23 Friday, 14:15

Invariance principle for the random Lorentz gas beyond the Boltzmann-Grad (or, Gallavotti-Spohn) limit - Part 1

Abstract

Let hard ball scatterers of radius r be placed in R^{d}, centred
at the points of a Poisson point process of intensity ρ. The volume
fraction r^{d}ρ is assumed to be sufficiently low so that with positive
probability the origin is not trapped in a finite domain fully surrounded by
scatterers. The Lorentz process is the trajectory of a point-like particle
starting from the origin with randomly oriented unit velocity subject to
elastic collisions with the fixed (infinite mass) scatterers. The question
of diffusive scaling limit of this process is one of the major open problems
in classical statistical physics.

Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r^{d-1}ρ→1 and the time scale is fixed, the Lorentz process (described
informally above) converges to a Markovian random flight process, with
independent exponentially distributed free flight times and Markovian
scatterings. It is essentially straightforward to see that taking a second
diffusive scaling limit (after the Gallavotti-Spohn limit) yields invariance
principle.

I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r^{d-1}ρ→1 (as in B-G) and simultaneously rescaling time by
T∼r^{-2+ε} we prove invariance principle (under diffusive
scaling) for the Lorentz trajectory. (Note that the B-G limit and diffusive
scaling are done simultaneously and not in sequel.) The proof is essentially
based on control of the effect of recollisions by probabilistic coupling
arguments. The main arguments are valid in d=3 but not in d=2.

Joint work (partly in progress) with Chris Lutsko (Bristol).

Gallavotti (1969) and Spohn (1978) proved that under the so-called Boltzmann-Grad limit, when r→0, ρ→∞ so that r

I will present new results going beyond the [Boltzmann-Grad / Gallavotti-Spohn] limit, in d=3: Letting r→0, ρ→∞ so that r

Joint work (partly in progress) with Chris Lutsko (Bristol).

2018.02.22 Thursday, 16:15