TU Budapest -- BME

talks in Fall semester 2017 -- előadásai a 2017 őszi félévben

Percolation games, ergodicity of probabilistic cellular automata, and the hard-core model

Abstract

Let each site of the square lattice Z^{2} be declared "closed" with
probability p, a "target" with probability q, and "open" with probability
1-p-q, where 0≤p≤p+q≤1. Different sites are independent.

Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site x to a site in {x+(0,1),x+(1,0)}. A player moving to a closed site loses immediately. A player moving to a target site wins immediately. Otherwise, the game continues.

Is there positive probability that the game is drawn with best play - i.e. that neither player can force a win? We show that this question is equivalent to the question of ergodicity of a certain elementary one-dimensional probabilistic cellular automaton (PCA). We prove that the PCA is ergodic whenever p+q>0, and correspondingly that the game on Z^{2} has
no draws.

On the other hand, we prove that for q=0 and p sufficiently small, the analogous game does exhibit draws on various directed graphs in dimension d at least 3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d-1. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Z^{d} for d at least 3, but here
our method encounters a fundamental obstacle.

If time permits, I will also mention similar games on undirected lattices, where there are a few related results but also many open questions.

This is joint work with Alexander Holroyd and Irene Marcovici.

Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site x to a site in {x+(0,1),x+(1,0)}. A player moving to a closed site loses immediately. A player moving to a target site wins immediately. Otherwise, the game continues.

Is there positive probability that the game is drawn with best play - i.e. that neither player can force a win? We show that this question is equivalent to the question of ergodicity of a certain elementary one-dimensional probabilistic cellular automaton (PCA). We prove that the PCA is ergodic whenever p+q>0, and correspondingly that the game on Z

On the other hand, we prove that for q=0 and p sufficiently small, the analogous game does exhibit draws on various directed graphs in dimension d at least 3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d-1. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Z

If time permits, I will also mention similar games on undirected lattices, where there are a few related results but also many open questions.

This is joint work with Alexander Holroyd and Irene Marcovici.

2018.01.18 Thursday, 16:15

Eigenvalues of random non-Hermitian matrices and randomly coupled differential equations

Abstract

We consider large random matrices with centered, independent entries but
possibly different variances and compute the limiting distribution of
eigenvalues. We then consider applications to long time asymptotics for
systems of critically coupled differential equations with random
coefficients.

2017.12.07 Thursday, 16:15

Error exponents for communication models with multiple codebooks and the capacity region of partly asynchronous multiple access channel

PhD public defense

szokatlan időpont / unusual time

szokatlan időpont / unusual time

2017.12.04 Monday, 15.00

Unusual place: BME J épület I. Em. 102..

Sign clusters of the Gaussian free field percolate on Z^d, d≥3

Abstract

We consider level set percolation for the Gaussian free field on the
Euclidean lattice in dimensions larger than or equal to three. It had
previously been shown by Bricmont, Lebowitz, and Maes that the critical
level is non-negative in any dimension and finite in dimension three.
Rodriguez and Sznitman have extended this result by proving that it is
finite in all dimensions, and positive in all large enough dimensions.

We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field.

This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).

We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field.

This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).

2017.11.23 Thursday, 16:15

Critical percolation on the Hamming graph

Abstract

Percolation on finite graphs is known to exhibit a phase transition similar
to the Erdős-Rényi Random Graph in presence of sufficiently weak geometry.
We focus on the Hamming graph H(d,n) (the cartesian product of d complete
graphs on n vertices each) when d is fixed and n→∞. We identify the
critical point p_{c}(d) at which such phase transition happens and we analyse
the structure of the largest connected components at criticality. We prove
that the scaling limit of component sizes is identical to the one for
critical Erdős-Rényi components, while the number of surplus edges is much
higher. These results are obtained coupling percolation to the trace of
branching random walks on the Hamming graph.

Based on joint work with Remco van der Hofstad, Frank den Hollander and Tim Hulshof.

Based on joint work with Remco van der Hofstad, Frank den Hollander and Tim Hulshof.

2017.11.09 Thursday, 16:15

Shock Fluctuations in TASEP

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with a
non-random initial condition that has a discontinuity (shock) in the
particle density. If one inserts a "second class particle" in the system,
it will follow the shock. For large time t, we show that the position of
the second class particle fluctuates on the t^{1/3} scale and we determine
its limiting law. Joint work with Patrik Ferrari and Promit Ghosal.

2017.11.02 Thursday, 16:15

The maximum of branching random walk in spatially random branching environment

Abstract

Branching branching random walk and Brownian motion have been the subject of
intensive research recently. We consider branching random walk and
investigate the effect of introducing a spatially random branching
environment. We are primarily interested in the position of the maximum
particle, for which we prove a CLT. Our result correspond, on an analytic
level, to a CLT for the front of the solutions to a randomized Fisher-KPP
equation, and also to a CLT for the parabolic Anderson model.

2017.10.26 Thursday, 16:15

Regular Decomposition: an information and graph theoretic approach to stochastic block models

Abstract

A method for compression of large graphs and matrices to a block structure is proposed. Szemerédi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We propose practical algorithms and provide theoretical results on the accuracy and consistency of the method.

2017.10.12 Thursday, 16:15

Finding patterns in Brownian motion

Abstract

We consider the problem of finding particular patterns in a realisation of a
two-sided standard Brownian motion. Examples include two-sided Skorohod
imbedding, the Brownian bridge and several other patterns, also in planar
Brownian motion. The key tool here are recent allocation results in Palm
theory.

2017.10.05 Thursday, 16:15