Papers
with Á. Backhausz, B. Gerencsér, M. Vizer:

Correlation bound for distant parts of factor of IID processes

Combinatorics, Probability and Computing, to appear.

pdf

with B. Gerencsér:

Mutual information decay for factors of IID

submitted.

pdf

with Á. Backhausz, B. Gerencsér:

Entropy inequalities for factors of IID

submitted.

pdf

Factors of IID on hypergraphs and entropy inequalities

in preparation.

Seminar talks
Invariant processes on infinite trees 
slides

LouvainlaNeuve, Belgium 
Abstract: In this talk we will demonstrate how studying certain invariant processes on infinite trees can help to answer questions about (large) finite graphs/networks (such as questions about the maximal size of an independent set or about the eigenvectors of the adjacency matrix). Our main focus will be the class of factor of IID processes (which are essentially factors of Bernoulli shifts from an ergodic theoretic point of view). We will also mention some entropy inequalities that played a central role in a couple of remarkable results recently. 
Entropy inequalities for factors of IID 

University of Toronto, Canada 
Abstract: This talk is concerned with certain invariant processes over infinite trees. Tha main focus will be on the class of factor of IID processes which have received a growing attention recently due to their connection to certain randomized local algorithms and random regular graphs. Entropy inequalities for these processes played a central role in a couple of remarkable results in the past few years. I will talk about how one can find and prove new entropy inequalities. 
Correlation bounds and oneended tail triviality for factors of IID on trees 
slides

Stochastics Seminar, Budapest University of Technology and Economics 
Abstract: We study factor of IID processes on the dregular 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 IID processes have trivial 1ended tails. 
On the eigenvectors of random regular graphs 

Graph Limits Seminar, Rényi Institute 
Abstract: There is much known about the spectrum of a random regular graph. When it comes to its eigenvectors, however, even (not too strong) delocalization results are hard to come by. In this talk we will survey what has been proved in the literature, and also mention some ambitious conjectures and potential approaches. 
The resolvent operator and the Stieltjes transform 

Graph Limits Seminar, Rényi Institute 
Abstract: Given a compactly supported measure on the real line (such
as the spectral measure of a bounded degree graph), its Stieltjes
transform is an analytic function on the complex upper halfplane.
This function can also be obtained directly from the adjacency
operator of the graph via the socalled resolvent operator. Stieltjes
transforms proved to be a useful tool to study the spectral properties
of (random) matrices/operators. In this talk we give a brief
introduction to the subject. 
Talks at student seminars
Randomizált lokális algoritmusok nagy gráfokon 
slides (partial)

Eötvös University Bolyai Kollégium 
Abstract (in Hungarian): Olyan párhuzamosítható algoritmusokat tekintünk gráfokon, melyeknél minden csúcs csupán egy fix sugarú környezetét "látja" és kizárólag ez alapján dönti el, hogyan "jár el". Olyan gráfoknál, ahol ezek a környezetek nagyrészt ugyanúgy néznek ki, "megtörhetjük a szimmetriát" azáltal, hogy első lépésként a csúcsokhoz színeket/címkéket rendelünk véletlenszerűen, és a randomizált lokális algoritmus már ezeket a címkéket is használhatja. Sok esetben hasznosnak bizonyul, ha az algoritmust a "limesz gráfon" vizsgáljuk. 
PacManek esete a hópehely görbével 
slides (Hungarian)

Eötvös University Bolyai Kollégium 
Abstract (in Hungarian): A Kochféle hópehely görbe az egyik legismertebb önhasonló halmaz. Az előadáson azt a meglepő tényt látjuk be, hogy ez a fraktál lefedhető tetszőlegesen kis összszélességű sávokkal. A bizonyításhoz PacManeket fogunk segítségül hívni. 
English version: the following similar talk was given a few months earlier. 
Tubenull sets 
slides
video

Central European University 
Abstract: A set in the plane is called 'tubenull' if it can be covered by strips of arbitrarily small total width. (A strip is the region between two parallel lines.) Any such set can appear as the set of divergence for the localization problem. It turns out that there are not many methods for proving that a given set is not tubenull, and even for very concrete sets, like the snowflake curve, it can be hard to decide whether they are tubenull or not. 
Workshop organization
Mini workshop on entropy inequalities for factors of IID,
August 2226, 2016, Siófok
Description. Entropy inequalities proved to be a key tool
in a couple of remarkable results recently:
the RahmanVirág result about the maximal size of a factor of IID
independent set on the regular tree and
the BackhauszSzegedy result on eigenvectors of random regular graphs.
The main focus of this mini workshop will be to investigate these inequalities,
look for further applications, and try to obtain new inequalities as well.
Another goal of the workshop is to study a series of papers by Lewis Bowen
in which he introduced the finvariant for free group actions
and the soficentropy for actions of sofic groups.
Bowen also showed that the finvariant is essentially a special case
of the the soficentropy which has the consequence that
the finvariant is nonnegative for factors of the Bernoulli shift.
This fact is particularly relevant for us
since it is equivalent to (a somewhat more general version of)
the edgevertex entropy inequality for factors over the free group.
Conference participation
 Groups, Graphs and Stochastic Processes, June 2126, 2015, BIRS, Banff, Canada.
link
 Spectrum of Random Graphs, January 48, 2016, CIRM, Luminy, France.
link
Upcoming events:
 2017 Colloquia in Combinatorics, Queen Mary and London School of Economics.
link
 "Graph limits, groups and stochastic processes" Summer School, Rényi Institute.
link
Public engagement
 Talk at a public event organized and hosted by the Hungarian Academy of Sciences.
I was invited to talk about the research I had done during the period of this fellowship
and about my experience with applying for and participating in a Marie SkłodowskaCurie grant.
The purpose of the event was multifold. On the one hand, it showcased the various research
carried out in Hungary under MSCA and ERC grants. Its main purpose was to popularize these grants
among Hungarian researchers and to provide advice and guidance on how to prepare a successful proposal.
link video (in Hungarian)
 Interview with Erinto
(an online mathematics portal published by the János Bolyai Mathematical Society)
Other activities
 Editorial work at one of the top combinatorial journals:
Managing Editor of Combinatorica (2015 August  present).
