## Final Report Summary - GRAPHCONVSTOCH (Graph Convergence and Stochastic Processes on Graphs)

Thermodynamical limit, a fundamental concept in statistical physics, is based on local convergence of boxes to Z^d, together with random objects on them. The main focus of statistical physics is on Euclidean lattices, trees, and their random subgraphs. For graph theorists and group theorists, other vertex transitive graphs are typically more interesting. The thermodynamical limiting operation in discrete mathematics is called Benjamini-Schramm limit. The study of graph parameters can lead to new results in statistical physics on the one hand, while the methods of statistical physics may gain applications in graph theory and computer science on the other. This interplay between statistical physics, probability and graph theory promises many exciting new results.

The proposal was mainly focused on the following areas:

1. Benjamini-Schramm convergence, unimodular random graphs, percolation models

2. Factors on Cayley graphs, unimodular graphs and of point processes

3. Processes on graphs and their limits

Timar spent the first ten years of his research carreer outside of Hungary. Receiving his PhD in the USA, as a student of Russell Lyons, he was “grown up” in the important and influential school of probabilists trademarked by Lyons and Peres. The Researcher in Charge, Abert, who is a group theorist with interest in ergodic theory, has arrived to similar questions as the three topics highlighted above, from directions different from Timar’s. The profile of Timar was perfectly suited and needed for Abert’s freshly growing research group, coming mainly from graph theory and algebra.

The main objective was to connect Timar's research to already existing research directions at the host institute in an organic way and to integrate Timar in the mathematical life of Hungary. In particular, one of the goals was a knowledge from Timar based on his expertize on percolation and factors of point processes towards the host, and in the other direction, to expand Timar’s knowledge on graph limits and measured equivalence relations. The expertise of Timar in factors of point processes, and that of the host in factor of iid processes resulted in synergies between the two fields. Timar spent particularly much time working with Abert, Pete and Beringer, all affiliated with the institute.

The most important results of the project are the proofs of indistinguishability for various classes of random spanning forests, and other models (line percolation). In the measurable group theory terminology these results can be translated as the ergodicity of the corresponding measurable group equivalence relation. These results answered questions and conjectures of Benjamini, Lyons, Peres and Schramm, and by Hilário and Sidoravicius. The type of weak insertion tolerance that was proved for the uniform and minimal spanning forests of a Cayley graph are expected to have further applications.

The study of the behavior of graph parameters under limit operations was one of the core elements of the current proposal. In a preprint with D. Beringer and G. Pete, a conjecture of Schramm on the locality of the percolation parameter is examined in the more general context of unimodular graphs. Since the famous conjecture is open even for Cayley graphs, the goal here was to see if the known direction of the proof can be extended to this more general class of graphs, and to identify the definition of critical probability that is most suitable for the study of locality. This latter problem is meaningful because, as it has turned out, the usual equivalent definitions of critical probability (Hammersley, Temperley,...) may be different in this context. The article containing the results of this research got submitted to an international journal.

The continuity of another graph parameter, the controllability number, has been extensively studied by network scientists. One of the goals was a rigorous mathematical study of the continuity of this parameter under limits, because the network theoretic research is rather simulation-based. In a preprint being written by Beringer and Timar, the almost sure continuity is proved for several families of scale-free (“real world”) networks. Speed of convergence is a research direction for the near future.

Itai Benjamini proposed the following question at one of the conferences that Timar attended during the fellowship. Is there a way to represent a nonamenable transitive graph, say, a regular tree, in R^3, as a partition into “nice” connected pieces, so that the pieces are indistinguishable? In other words, is there a 3-dimensional isometry-invariant “map” of indistinguishable regions whose adjacency graph is the 3-regular tree? The answer was expected to be negative because of the known case of R^2, and because of the intuition that amenability should be an obstacle. However, Timar came up with a construction that shows the possibility of such a representation. Along the way he proved that any amenable Cayley graph can be invariantly embedded in R^d (d>2). The results were written up and submitted. A related work was started with Benjamini, to investigate various embeddability questions, a paper “Invariant embeddings of graphs” is being written.

In mathematics, the time needed from a preprint stage to actual publication of a result is notoriously long. One of an older results of Timar had the last polishing and got published during the period of the fellowship, two others also got published in this time interval, while having been written and final version submitted before the beginning of the fellowship. Finally, the preprints produced during the time of the fellowship are still being reviewed by journals. They are available on the project website and on arxiv.org.

A paper in the process of writing solves questions of Bandyopadhyay and coauthors on random walk in random environment. Namely, the speed of the random walk on a regular tree is determined for certain general classes of environments.

Timar has been an active an enthusiastic member of the Hungarian mathematical community. Besides giving talks at many Hungarian universities, he also gave popular talks to students. He wrote a paper to a popular science journal (under review) in Hungarian, to present the Benford law and correct a related research result in social sciences that has become widely known. Timar has been the advisor of a PhD student, Dorottya Beringer, who is a member of the research group of Abert. Timar gave several introductory talks to the group of younger researchers in Abert’s group, and he was conducting a research seminar for undergraduates.

To summarize, the project successfully fulfilled its goals both scientifically and from the transfer of knowledge and dissemination points of view.

For further details, see the project homepage: http://www.renyi.hu/GraphConvStoch.html

The proposal was mainly focused on the following areas:

1. Benjamini-Schramm convergence, unimodular random graphs, percolation models

2. Factors on Cayley graphs, unimodular graphs and of point processes

3. Processes on graphs and their limits

Timar spent the first ten years of his research carreer outside of Hungary. Receiving his PhD in the USA, as a student of Russell Lyons, he was “grown up” in the important and influential school of probabilists trademarked by Lyons and Peres. The Researcher in Charge, Abert, who is a group theorist with interest in ergodic theory, has arrived to similar questions as the three topics highlighted above, from directions different from Timar’s. The profile of Timar was perfectly suited and needed for Abert’s freshly growing research group, coming mainly from graph theory and algebra.

The main objective was to connect Timar's research to already existing research directions at the host institute in an organic way and to integrate Timar in the mathematical life of Hungary. In particular, one of the goals was a knowledge from Timar based on his expertize on percolation and factors of point processes towards the host, and in the other direction, to expand Timar’s knowledge on graph limits and measured equivalence relations. The expertise of Timar in factors of point processes, and that of the host in factor of iid processes resulted in synergies between the two fields. Timar spent particularly much time working with Abert, Pete and Beringer, all affiliated with the institute.

The most important results of the project are the proofs of indistinguishability for various classes of random spanning forests, and other models (line percolation). In the measurable group theory terminology these results can be translated as the ergodicity of the corresponding measurable group equivalence relation. These results answered questions and conjectures of Benjamini, Lyons, Peres and Schramm, and by Hilário and Sidoravicius. The type of weak insertion tolerance that was proved for the uniform and minimal spanning forests of a Cayley graph are expected to have further applications.

The study of the behavior of graph parameters under limit operations was one of the core elements of the current proposal. In a preprint with D. Beringer and G. Pete, a conjecture of Schramm on the locality of the percolation parameter is examined in the more general context of unimodular graphs. Since the famous conjecture is open even for Cayley graphs, the goal here was to see if the known direction of the proof can be extended to this more general class of graphs, and to identify the definition of critical probability that is most suitable for the study of locality. This latter problem is meaningful because, as it has turned out, the usual equivalent definitions of critical probability (Hammersley, Temperley,...) may be different in this context. The article containing the results of this research got submitted to an international journal.

The continuity of another graph parameter, the controllability number, has been extensively studied by network scientists. One of the goals was a rigorous mathematical study of the continuity of this parameter under limits, because the network theoretic research is rather simulation-based. In a preprint being written by Beringer and Timar, the almost sure continuity is proved for several families of scale-free (“real world”) networks. Speed of convergence is a research direction for the near future.

Itai Benjamini proposed the following question at one of the conferences that Timar attended during the fellowship. Is there a way to represent a nonamenable transitive graph, say, a regular tree, in R^3, as a partition into “nice” connected pieces, so that the pieces are indistinguishable? In other words, is there a 3-dimensional isometry-invariant “map” of indistinguishable regions whose adjacency graph is the 3-regular tree? The answer was expected to be negative because of the known case of R^2, and because of the intuition that amenability should be an obstacle. However, Timar came up with a construction that shows the possibility of such a representation. Along the way he proved that any amenable Cayley graph can be invariantly embedded in R^d (d>2). The results were written up and submitted. A related work was started with Benjamini, to investigate various embeddability questions, a paper “Invariant embeddings of graphs” is being written.

In mathematics, the time needed from a preprint stage to actual publication of a result is notoriously long. One of an older results of Timar had the last polishing and got published during the period of the fellowship, two others also got published in this time interval, while having been written and final version submitted before the beginning of the fellowship. Finally, the preprints produced during the time of the fellowship are still being reviewed by journals. They are available on the project website and on arxiv.org.

A paper in the process of writing solves questions of Bandyopadhyay and coauthors on random walk in random environment. Namely, the speed of the random walk on a regular tree is determined for certain general classes of environments.

Timar has been an active an enthusiastic member of the Hungarian mathematical community. Besides giving talks at many Hungarian universities, he also gave popular talks to students. He wrote a paper to a popular science journal (under review) in Hungarian, to present the Benford law and correct a related research result in social sciences that has become widely known. Timar has been the advisor of a PhD student, Dorottya Beringer, who is a member of the research group of Abert. Timar gave several introductory talks to the group of younger researchers in Abert’s group, and he was conducting a research seminar for undergraduates.

To summarize, the project successfully fulfilled its goals both scientifically and from the transfer of knowledge and dissemination points of view.

For further details, see the project homepage: http://www.renyi.hu/GraphConvStoch.html