Papers
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with Á. Backhausz, B. Gerencsér, M. Vizer:
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Correlation bound for distant parts of factor of IID processes
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Combinatorics, Probability and Computing, to appear.
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pdf
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with B. Gerencsér:
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Mutual information decay for factors of IID
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Ergodic Theory and Dynamical Systems, to appear.
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pdf
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with Á. Backhausz, B. Gerencsér:
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Entropy inequalities for factors of IID
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submitted.
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pdf
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Factors of IID on hypergraphs and entropy inequalities
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in preparation.
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Seminar talks
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Invariant processes on infinite trees |
slides
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Louvain-la-Neuve, Belgium |
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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. |
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Entropy inequalities for factors of IID |
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University of Toronto, Canada |
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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. |
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Correlation bounds and one-ended tail triviality for factors of IID on trees |
slides
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Stochastics Seminar, Budapest University of Technology and Economics |
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Abstract: We study factor of IID processes on the d-regular 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 1-ended tails. |
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On the eigenvectors of random regular graphs |
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Graph Limits Seminar, Rényi Institute |
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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. |
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The resolvent operator and the Stieltjes transform |
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Graph Limits Seminar, Rényi Institute |
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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 half-plane.
This function can also be obtained directly from the adjacency
operator of the graph via the so-called 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
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Randomizált lokális algoritmusok nagy gráfokon |
slides (partial)
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Eötvös University Bolyai Kollégium |
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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. |
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Pac-Man-ek esete a hópehely görbével |
slides (Hungarian)
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Eötvös University Bolyai Kollégium |
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Abstract (in Hungarian): A Koch-féle hópehely görbe az egyik legismertebb önhasonló halmaz. Az előadáson azt a meglepő tényt látjuk be, hogy ez a fraktál lefedhető tetszőlegesen kis össz-szélességű sávokkal. A bizonyításhoz Pac-Man-eket fogunk segítségül hívni. |
English version: the following similar talk was given a few months earlier. |
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Tube-null sets |
slides
video
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Central European University |
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Abstract: A set in the plane is called 'tube-null' 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 tube-null, and even for very concrete sets, like the snowflake curve, it can be hard to decide whether they are tube-null or not. |
Workshop organization
Mini workshop on entropy inequalities for factors of IID,
August 22-26, 2016, Siófok
Description. Entropy inequalities proved to be a key tool
in a couple of remarkable results recently:
the Rahman-Virág result about the maximal size of a factor of IID
independent set on the regular tree and
the Backhausz-Szegedy 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 f-invariant for free group actions
and the sofic-entropy for actions of sofic groups.
Bowen also showed that the f-invariant is essentially a special case
of the the sofic-entropy which has the consequence that
the f-invariant is non-negative 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 edge-vertex entropy inequality for factors over the free group.
Conference participation
- Groups, Graphs and Stochastic Processes, June 21-26, 2015, BIRS, Banff, Canada.
link
- Spectrum of Random Graphs, January 4-8, 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 Skłodowska-Curie 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).
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Website
