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Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice

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Daniel Ahlberg

Probability · Submitted 12 Jul 2011

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice.

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arXiv ID1107.2280 (version 4)
Pith IDIARUCBJJ, Pith's permanent record of this paper and its review
CitationDaniel Ahlberg. "Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice." arXiv:1107.2280 (2011). https://pith.science/paper/IARUCBJJ
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,693 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.42 · mentions code or data availability 0.02.

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arXiv listing abstract v4. Results and the main claim are underlined; .

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. background 1.00 0.04 0.04 asserted 1.00
s1 Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. contribution 0.84 0.67 0.07 asserted 0.70
s2 In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. result 0.97 0.93 0.72 asserted 1.00
s3 Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment. contribution 0.99 0.94 0.16 asserted 0.99

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The paper
Original abstract

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.

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2 arXiv papers cite this in 3 sentences

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Compares with it 1 paper
  • A similar characterization of deviations away from the time constant as the one presented here has in parallel been derived for first-passage percolation of cone-like subgraphs of the $^d$ lattice by the same author [ref] .

    A Hsu-Robbins-Erd\H{o}s strong law in first-passage percolation · Large deviation estimates · 0.92

What it cites · 9 sentences without a confident label

Builds on 1 paper
  • Note that deviations below and above the time constant may easily be treated separately via the identity [equation] Since travel times on $$ are bounded from below by travel times on the lattice, we will be able to use the fact (from [ref] ) that Theorem [ref] holds for the $^d$ lattice to obtain an estimate on deviations below the time constant.

    A Hsu-Robbins-Erd\H{o}s strong law in first-passage percolation · A large deviations estimate for cones · 0.80

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