Correlation Bounds for Distant Parts of Factor of IID Processes

Backhausz, Ágnes; Gerencsér, Balázs; Harangi, Viktor; Vizer, Máté

doi:10.1017/s0963548317000360

Cited by 5 publications

(9 citation statements)

References 40 publications

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“…(Because of the Aut(T d )-invariance the distribution of X v is the same for each vertex v. Similarly, the joint distribution of (X u , X v ) is the same for any edge uv.) Rahman and Virág proved (1) in a special setting [16]. A full and concise proof was given by Backhausz and Szegedy in [2]; see also [15].…”

Section: Introductionmentioning

confidence: 90%

“…on R V (T d ) so that (9) can be applied. The second ingredient is the next lemma from [1] which tells us that the same bound holds if we take different real-valued functions of X u and X v .…”

Section: The Rate Of Decay For a Fixed Processmentioning

confidence: 99%

“…Proof. The detailed proof can be found in [1,Lemma 3.2]. We include a sketch here for the sake of completeness.…”

Section: The Rate Of Decay For a Fixed Processmentioning

confidence: 99%

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Mutual information decay for factors of i.i.d.

Gerencsér

Harangi²

2018

Ergod. Th. Dynam. Sys.

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This paper is concerned with factors of independent and identically distributed processes on the $d$ -regular tree for $d\geq 3$ . We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$ ), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$ , of order $(d-1)^{-k/2}$ . Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$ .

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Section: Introductionmentioning

confidence: 90%

Section: The Rate Of Decay For a Fixed Processmentioning

confidence: 99%

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Mutual information decay for factors of i.i.d.

Gerencsér

Harangi²

2018

Ergod. Th. Dynam. Sys.

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“…Regardless, there are many techniques, such as entropy inequality [BGH19] or correlation decay [BSV15], and particular results such as the aforementioned perfect matching problem [LN11] that provide lower and upper bounds, respectively, in our setting as well. We refer the reader to [Lyo17,BGHV18] for a comprehensive summary of the field.…”

Section: Distributed Computingmentioning

confidence: 99%

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

Brandt,

Chang,

Grebík

et al. 2021

Preprint

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We study connections between three different fields: distributed local algorithms, finitary factors of iid processes, and descriptive combinatorics. We focus on two central questions: Can we apply techniques from one of the areas to obtain results in another? Can we show that complexity classes coming from different areas contain precisely the same problems? We give an affirmative answer to both questions in the context of local problems on regular trees:

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“…The other important special case is when T is the universal cover of a finite connected simple graph G and Γ = Γ ϕ is the group of covering transformations for a covering ϕ : T → G. In this case it holds that for anyv 1…”

Section: Factors Of Iidmentioning

confidence: 99%

Entropy inequalities for factors of IID

2018

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This paper is concerned with certain invariant random processes (called factors of IID) on infinite trees. Given such a process, one can assign entropies to different finite subgraphs of the tree. There are linear inequalities between these entropies that hold for any factor of IID process (e.g. "edge versus vertex" or "star versus edge"). These inequalities turned out to be very useful: they have several applications already, the most recent one is the Backhausz-Szegedy result on the eigenvectors of random regular graphs.We present new entropy inequalities in this paper. In fact, our approach provides a general "recipe" for how to find and prove such inequalities. Our key tool is a generalization of the edge-vertex inequality for a broader class of factor processes with fewer symmetries.2010 Mathematics Subject Classification. 37A35, 60K35, 37A50, 05E18.

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Correlation Bounds for Distant Parts of Factor of IID Processes

Cited by 5 publications

References 40 publications

Mutual information decay for factors of i.i.d.

Mutual information decay for factors of i.i.d.

Local Problems on Trees from the Perspectives of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics

Entropy inequalities for factors of IID

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