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Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model

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Omar Boukhadra

Probability · Submitted 26 Jul 2009

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

The authors' main claim We show that for $\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order.

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arXiv ID0907.4525 (version 3)
Pith IDVJIZRDWG, Pith's permanent record of this paper and its review
CitationOmar Boukhadra. "Standard Spectral Dimension for the Polynomial Lower Tail Random Conductances model." arXiv:0907.4525 (2009). https://pith.science/paper/VJIZRDWG
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,677 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.03 · states quantitative result 0.82 · compares with prior results 0.68 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $\gamma$ near 0. We are interested in estimating the quenched decay of the return probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for $\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. As an expected concequence, the same result holds for the discrete-time case.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $\gamma$ near 0. background 0.50 0.29 0.03 asserted 0.83
s1 We are interested in estimating the quenched decay of the return probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. background 0.61 0.26 0.03 asserted 0.49
s2 We show that for $\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. result 1.00 0.92 0.78 asserted 1.00
s3 As an expected concequence, the same result holds for the discrete-time case. interpretation 0.76 0.55 0.09 asserted 0.88

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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  • references and citing sentences
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  • C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study models of continuous-time, symmetric, $\Z^{d}$-valued random walks in random environments, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$ with a power law with an exponent $\gamma$ near 0. We are interested in estimating the quenched decay of the return probability $P_\omega^{t}(0,0)$, as $t$ tends to $+\infty$. We show that for $\gamma> \frac{d}{2}$, the standard bound turns out to be of the correct logarithmic order. As an expected concequence, the same result holds for the discrete-time case.

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