Uniform spanning forests

By Itai Benjamini, Russell Lyons, Yuval Peres, Oded Schramm. 2001

271 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b973f611a54f6cca51f4a4b66c7247c5267ed57d

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Mathematical aspects of the abelian sandpile model (2005, background): “In [ 4 ], theorem 10.1, it is proved that each component of the USF has one end.”
The components of the Wired Spanning Forest are recurrent (2003, methodology): “Following [ BLPS ], we will call the the energy-minimizing “free” unit flow free current and the energy-minimizing “wired” flow wired current and denote them by I e F and I e”
Random planar curves and Schramm-Loewner evolutions (2003, methodology): “on 9.3 (Wilson’s algorithm) The law of A m is the uniform spanning tree measure. Note that this algorithm yields a natural extension of uniform spanning trees (or forests) in inﬁnite graphs (see e.g. [20] and the references therein for more on this subject). Proof. One can derive this result using t”
A measurable-group-theoretic solution to von Neumann’s problem (2007, methodology): “Finally, the fact that the clusters of π(x), and hence those of F2, have infinitely many ends follows, e.g., from [ 2, Theorem 13.7 ].”
Determinantal Processes and Independence (2005, unknown): “We can assume that K is trace class, since it suffices to construct a determinantal process on compact subsets of Λ with kernel the restriction of K. Sufficiency: If K is a projection operator, this is precisely Lemma 21.”
Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models (2004, background): “Given a quantity f(Λ) defined for all (or all sufficiently large) finite subsets Λ of Zd we say that lim Λ↗Zd f(Λ) = a, if given any ε > 0 there is a finite Λ0 ⊂ Zd such that for all finite Λ ⊃ Λ0 we have |f(Λ)− a| < ε.”
Ends in Uniform Spanning Forests (2007, background): “see Lyons (1998) for a survey and Benjamini, Lyons, Peres, and Schramm (2001) , later”
Hyperbolic and Parabolic Unimodular Random Maps (2016, methodology): “Benjamini, Lyons, Peres and Schramm [13] also relate analytic properties of G to the WUSF and FUSF of G. Theorem 5.8 ([13]; Theorem 1, equivalence of items (12) (11)).”
Determinantal point processes (2018, methodology): “is the tail of e, −1, if vis the head of e, 0, otherwise. Then [K(e,f)]e,f∈X is the matrix of the orthogonal projection operator with image Span{a(v)}, where vvaries over all vertices of G, see e. g. [Ben01]. Theorem 8.1 P is a determinantal point process with correlation kernel K. Theorem 8.1 was p”
Loop-erased walks and total positivity (2000, result): “This observation was related to me by several experts, including Y. Peres and O. Schramm, who based it on the setup of [ 1 ].”
Processes on Unimodular Random Networks (2006, background): “Another example concerns monotone coupling of the wired and free uniform spanning forests (whose definitions are given below in Section 7). This question was raised in Benjamini, Lyons, Peres, and Schramm (2001) , hereinafter referred to as BLPS (2001); a partial answer was given by Bowen (2004).”
The Random-Cluster Model (2002, background): “The theory of random spanning trees is beautiful in its own right (see [14]), and is linked in an important way to the emerging field of stochastic growth processes of ‘stochastic Löwner evolution’ (SLE) type (see [105, 124]), to which we return in Section 6.”