Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

By Assaf Naor, Yuval Peres, Oded Schramm, Scott Sheffield. 2004

124 citations recorded by the source corpus.

A metric space X has Markov type 2, if for any reversible flnite-state Markov chain fZtg (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfles E(D 2) • K 2 tE(D 2) for some K = K(X) 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < 1, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension deflned on all of Lp.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eea38b54276dd45fd98d4f5e034fe02cefe6f20f

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Complex Interpolation between Hilbert, Banach and Operator spaces (2008, background): “ng graphs (cf. e.g. [45]). We refer the reader to the web site http://kam.mﬀ.cuni.cz/∼matousek/metrop.ps for a list of problems (in particular Linial’s contribution) related to this section. See also [53] for a similar theme. Let Gbe a ﬁnite set equipped with a graph structure, so that Gis the verte”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, unknown): “1 in [53] Lemma 5.”
Comparison of Metric Spectral Gaps (2013, methodology): “Below we will show how a combination of [80] and Theorem 1.”
Embeddings of Discrete Groups and the Speed of Random Walks (2007, methodology): “For p > 2 it was shown in [28] that Lp has Markov type 2 with constants O (√ p ) .”
Vector‐valued optimal Lipschitz extensions (2010, background): “We remark that the general problem of finding Lipschitz extensions of a Lipschitz function from a subset Y of a metric space X to another metric space Z has been thoroughly studied, with diverse applications in mathematics, computer science, and engineering (see the books [20, 3], as well as the int”
Speed of random walks, isoperimetry and compression of finitely generated groups (2015, unknown): “1215/00127094-2011-002 [69] Naor, Assaf; Peres, Yuval; Schramm, Oded; Sheffield, Scott, Markov chains in smooth {B}anach spaces and {G}romovhyperbolic metric spaces, Duke Math.”
Lp compression, traveling salesmen, and stable walks (2009, methodology): “Secondly the embedding was given in [44] without a detailed proof of its compression bounds, and since it is based on a different and simpler approach than our proof of Theorem 3.1 it is worthwhile to explain it here.”
Probability on Trees and Networks (2017, background): “26), but on transitive graphs, it is quite restricted, as shown by the following theorem of Newman and Schulman (1981). The proof is a beautiful combination of insertion tolerance and ergodicity.”