Secondary Novikov-Shubin invariants of groups and quasi-isometry

Oguni, Shin-ichi

doi:10.2969/jmsj/1180135508

“…The proof of the preceding theorem follows from the proof of Theorem 4.4 (parts (ii) and (iii)) in [89]. As an input we need the following inequalities for µ (2n) (ι), the return probability of the simple random walk after 2n steps: a 1 e −a2n 1/3 ≤ µ (2n) (ι) ≤ A 1 e −A2n 1/3 , for some positive constants a i , A i , i = 1, 2, and all positive integers n. For the reference see Theorem 15.15 in [115].…”

Section: The Lamplighter Groupmentioning

confidence: 99%

“…Bipartiteness is not only sufficient, but also necessary for the mentioned symmetry. Claim 4.5 from [89] shows that on amenable non-bipartite graphs 2k is not in the spectrum of ∆, thus the symmetry fails. Now we introduce percolation Laplacians associated to the configuration ω as bounded, selfadjoint operators on ℓ 2 (V (ω)).…”

Section: Percolation Hamiltonians On Cayley Graphsmentioning

confidence: 99%

“…The quantity (16) may be called a Lifshitz exponent or maybe more appropriately secondary Novikov-Shubin invariant of the Laplacian, using the terminology of [89]. The proof of the preceding theorem follows from the proof of Theorem 4.4 (parts (ii) and (iii)) in [89]. As an input we need the following inequalities for µ (2n) (ι), the return probability of the simple random walk after 2n steps: , for all E small enough.…”

Section: The Lamplighter Groupmentioning

confidence: 99%

“…Bipartiteness is not only sufficient, but also necessary for the mentioned symmetry. Claim 4.5 from [89] shows that on amenable non-bipartite graphs 2k is not in the spectrum of ∆, thus the symmetry fails.…”

Section: Consider An Arbitrary Subgraphmentioning

confidence: 99%

“…It establishes a Lifshitz-type asymptotics in the sense that (16) lim Of course the underlying operator is not random, but the IDS is exponentially thin at the bottom of the spectrum as it is the case for spectral fluctuation boundaries of random operators. The quantity ( 16) may be called a Lifshitz exponent or maybe more appropriately secondary Novikov-Shubin invariant of the Laplacian, using the terminology of [89].…”

Section: The Lamplighter Groupmentioning

confidence: 99%

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Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

Autunović

¹

,

Veselić

²

Methods of Spectral Analysis in Mathematical Physics

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Abstract. In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum.The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem.

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Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

Antunović¹,

Veselić²

2009

Journal de Mathématiques Pures et Appliquées

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We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models

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Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

Antunović,

Veselić

2007

Preprint

0

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We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.

show abstract

Secondary Novikov-Shubin invariants of groups and quasi-isometry

Cited by 3 publications

References 9 publications

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

Spectral Asymptotics of Percolation Hamiltonians on Amenable Cayley Graphs

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

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