The Objective Method: Probabilistic Combinatorial Optimization and Local Weak Convergence

By David J. Aldous, J. Michael Steele. 2004

478 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9850cc6b66e9d2d3cda43b7529ef270ebee1329f

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Unimodular random trees (2012, background): “A special form of the mass-transport principle was considered, in different language, by Aldous and Steele [3].”
Spectra of Large Random Trees (2009, methodology): “In the literature, the convergence of the local structure around a uniformly chosen vertex of Tn to the structure around the root for some limiting random sin-tree is an instance of what has been termed “local weak convergence” by Aldous, see [3].”
A large deviation principle for Wigner matrices without Gaussian tails (2012, methodology): “In this section, we start by adapting to our setting the notion of local weak convergence of rooted networks, introduced in [6], [2], and [1].”
Large deviations of empirical neighborhood distribution in sparse random graphs (2013, methodology): “We formulate the problem within the theory of local weak convergence of graph sequences that was recently introduced by Benjamini and Schramm [4] and Aldous and Steele [2].”
On exchangeable random variables and the statistics of large graphs and hypergraphs (2008, methodology): “Perhaps the simplest instance of the extraction of a limit object as a measure defined directly on a space of isomorphism classes of infinite models occurs in the work on limit objects for various classes of sparse rooted graph (see, in particular, the papers of Aldous and Steele [5], Aldous and Lyo”
Finding blocks and other patterns in a random coloring of Z (2006, methodology): “The tree structure was essential in [AS03] for certain computations and the approach does not extend directly to graphs like G(n, c/n) with c > 1, where the convenience of a tree structure is lost due to the presence of a giant component.”
A survey of max-type recursive distributional equations (2004, background): “re is that there is an n → ∞ “local weak limit” structure called the PWIT (Poisson weighted inﬁnite tree), which describes the geometry of the space as seen from a ﬁxed reference point. In brief (see [11] for more details) consider a Poisson point process (76) 0<ξ 1<ξ 2<ξ 3<··· 48 D. J. ”
The local weak limit of the minimum spanning tree of the complete graph (2013, methodology): “As is the case for many random combinatorial optimization problems on the complete graph, the local weak limit of Mn is naturally described in terms of the Poisson-weighted infinite tree (PWIT) of Aldous and Steele [5].”
The Radial Spanning Tree of a Poisson Point Process (2007, methodology): “This theorem has to be related to the convergence of geometric graphs as it is defined for the objective method (refer to [1]).”
Planar stochastic hyperbolic triangulations (2016, unknown): “[3, 5, 15, 17, 18, 29] and the references therein.”
Gaussian Limits for Random Measures in Geometric Probability (2005, background): “that HG φ is strogly stabilizing and satisﬁes the bounded moments condition. (i) If ξG φ satisﬁes (2.20) for p=4, if G is polynomially stabilizing, then for all τ >0, lim λ→∞ Var[HG φ (λ 1/d(P λτ ∩[0,1]d))] λ =VG φ (τ)·τ and for all f ∈C(A), lim n→∞ Var[hf,ρG φ,ni] n = Z A f2(x)V G φ (κ(x))κ(x)dx”
The functional equation of the smoothing transform (2009, unknown): “The Annals of Probability 2012, Vol. 40, No. 5, 2069–2105 DOI: 10.1214/11-AOP670 © Institute of Mathematical Statistics, 2012”
Laws of large numbers in stochastic geometry with statistical applications (2007, methodology): “We use a version of the ‘objective method’ [1,21], whereby convergence in distribution (denoted D −→) for a functional defined on a sequence of finite probabilistic objects (in this case, rescaled marked point processes) is established by showing that these probabilistic objects themselves converge ”