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Correlation Bounds for Distant Parts of Factor of IID Processes

Published online by Cambridge University Press:  01 August 2017

ÁGNES BACKHAUSZ ,
BALÁZS GERENCSÉR ,
VIKTOR HARANGI  and
MÁTÉ VIZER
ÁGNES BACKHAUSZ
Affiliation:
Department of Probability and Statistics, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (e-mail: agnes@cs.elte.hu) MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053 Budapest, Hungary (e-mail: gerencser.balazs@renyi.mta.hu, harangi@renyi.hu, vizermate@gmail.com)
BALÁZS GERENCSÉR
Affiliation:
Department of Probability and Statistics, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary (e-mail: agnes@cs.elte.hu) MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053 Budapest, Hungary (e-mail: gerencser.balazs@renyi.mta.hu, harangi@renyi.hu, vizermate@gmail.com)
VIKTOR HARANGI
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053 Budapest, Hungary (e-mail: gerencser.balazs@renyi.mta.hu, harangi@renyi.hu, vizermate@gmail.com)
MÁTÉ VIZER
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053 Budapest, Hungary (e-mail: gerencser.balazs@renyi.mta.hu, harangi@renyi.hu, vizermate@gmail.com)
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Abstract

We study factor of i.i.d. processes on the d-regular tree for d ≥ 3. 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, any functions of the two parts have correlation at most $k(d-1) / (\sqrt{d-1})^k$, where k denotes the distance between the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.

Keywords

Primary 60K35Secondary 37A50
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 1 , January 2018 , pp. 1 - 20
Copyright
Copyright © Cambridge University Press 2017 

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References

[1] Alon, N., Benjamini, I., Lubetzky, E. and Sodin, S. (2007) Non-backtracking random walks mix faster. Commun. Contemp. Math. 9 585–603.Google Scholar
[2] Angel, O., Friedman, J. and Hoory, S. (2015) The non-backtracking spectrum of the universal cover of a graph. Trans. Amer. Math. Soc. 367 4287–4318.Google Scholar
[3] Backhausz, Á. and Szegedy, B. (2014) On large girth regular graphs and random processes on trees. arXiv:1406.4420 Google Scholar
[4] Backhausz, Á. and Virág, B. (2015) Spectral measures of factor of i.i.d. processes on vertex-transitive graphs. Ann. Inst. Henri Poincaré Probab. Stat., to appear. arXiv:1505.07412 Google Scholar
[5] Backhausz, Á., Szegedy, B. and Virág, B. (2015) Ramanujan graphings and correlation decay in local algorithms. Random Struct. Alg. 47 424–435.CrossRefGoogle Scholar
[6] Ball, K. (2005) Factors of independent and identically distributed processes with non-amenable group actions. Ergodic Theory Dyn. Syst. 25 711–730.Google Scholar
[7] Ben-Hamou, A. and Salez, J. (2017) Cutoff for non-backtracking random walks on sparse random graphs. Ann. Probab., 45, no. 3, 1752–1770.CrossRefGoogle Scholar
[8] Bordenave, C. (2015) A new proof of Friedman's second eigenvalue theorem and its extension to random lifts. arXiv:1502.04482 Google Scholar
[9] Bowen, L. (2010) The ergodic theory of free group actions: Entropy and the f-invariant. Groups Geom. Dyn. 4 419–432.CrossRefGoogle Scholar
[10] Bowen, L. (2012) Sofic entropy and amenable groups. Ergodic Theory Dynam. Systems 32 427–466.CrossRefGoogle Scholar
[11] Conley, C. T., Marks, A. S. and Tucker-Drob, R. (2016) Brooks's theorem for measurable colorings. Forum of Mathematics, Sigma, 4 e16. DOI: https://doi.org/10.1017/fms.2016.14.CrossRefGoogle Scholar
[12] Csóka, E. (2016) Independent sets and cuts in large-girth regular graphs. arXiv:1602.02747 Google Scholar
[13] Csóka, E. and Lippner, G. (2017) Invariant random matchings in Cayley graphs. Groups, Geometry and Dynamics, 11 211–243.Google Scholar
[14] Csóka, E., Gerencsér, B., Harangi, V. and Virág, B. (2015) Invariant Gaussian processes and independent sets on regular graphs of large girth. Random Struct. Alg. 47 284–303.Google Scholar
[15] Friedman, J. (2008) A Proof of Alon's Second Eigenvalue Conjecture and Related Problems, Vol. 195, no. 910 of Memoirs of the American Mathematical Society, AMS.Google Scholar
[16] Gaboriau, D. and Lyons, R. (2009) A measurable-group-theoretic solution to von Neumann's problem. Invent. Math. 177 533–540.Google Scholar
[17] Gamarnik, D. and Sudan, M. (2014) Limits of local algorithms over sparse random graphs. In ITCS 2014: Proc. 5th Conference on Innovations in Theoretical Computer Science, ACM, pp. 369–376.Google Scholar
[18] Harangi, V. and Virág, B. (2015) Independence ratio and random eigenvectors in transitive graphs. Ann. Probab. 43 2810–2840.Google Scholar
[19] Hoppen, C. and Wormald, N. Local algorithms, regular graphs of large girth, and random regular graphs. Combinatorica, to appear. arXiv:1308.0266 Google Scholar
[20] Kechris, A. S. and Tsankov, T. (2008) Amenable actions and almost invariant sets. Proc. Amer. Math. Soc. 136 687–697.Google Scholar
[21] Kempton, M. (2016) Non-backtracking random walks and a weighted Ihara's theorem. Open J. Discrete Math. 6 207–226.Google Scholar
[22] Kerr, D. and Li, H. (2013) Soficity, amenability, and dynamical entropy. Amer. J. Math. 135 721–761.CrossRefGoogle Scholar
[23] Kun, G. (2013) Expanders have a spanning Lipschitz subgraph with large girth. arXiv:1303.4982 Google Scholar
[24] Lyons, R. (2017) Factors of IID on trees. Combin. Probab. Comput. 26 285–300.CrossRefGoogle Scholar
[25] Lyons, R. and Nazarov, F. (2011) Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32 1115–1125.Google Scholar
[26] Ornstein, D. S. and Weiss, B. (1987) Entropy and isomorphism theorems for actions of amenable groups. J. Analyse Math. 48 1–141.Google Scholar
[27] Pemantle, R. (1992) Automorphism invariant measures on trees. Ann. Probab. 20 1549–1566.Google Scholar
[28] Puder, D. (2015) Expansion of random graphs: New proofs, new results. Invent. Math. 201 845–908.Google Scholar
[29] Rahman, M. (2016) Factor of iid percolation on trees. SIAM J. Discrete Math. 30 2217–2242.CrossRefGoogle Scholar
[30] Rahman, M. and Virág, B. (2017) Local algorithms for independent sets are half-optimal. Ann. Probab., 45, no. 3, 1543–1577.Google Scholar
[31] Rokhlin, V. A. and Sinai, Y. G. (1961) Construction and properties of invariant measurable divisions. Doklady Akademii Nauk SSSR 141 1038–1041.Google Scholar
[32] Seward, B. (2016) Weak containment and Rokhlin entropy. arXiv:1602.06680 Google Scholar