Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

Disertori, Margherita; Merkl, Franz; Rolles, Silke W. W.

doi:10.30757/alea.v16-07

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…For the analysis of these random Schrödinger operators, certain martingales derived from the spin variables of the H 2|2 -model play an essential role. The authors generalized these martingales in Disertori et al (2017) and described supersymmetric variants of them in Disertori et al (2019).…”

Section: Introductionmentioning

confidence: 99%

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

Disertori¹,

Merkl²,

Rolles³

2022

ALEA

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We study the non-linear supersymmetric hyperbolic sigma model H 2|2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2|2 -model; its size is logarithmic in the size of the original model.

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Section: Introductionmentioning

confidence: 99%

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

Disertori¹,

Merkl²,

Rolles³

2022

ALEA

Self Cite

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show abstract

“…For the analysis of these random Schrödinger operators, certain martingales derived from the spin variables of the H 2|2 -model play an essential role. These martingales are generalized in [DMR17] and supersymmetric variants of them are described in [DMR19].…”

Section: Introduction 1history Of the H 2|2 -Model And Related Modelsmentioning

confidence: 99%

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

Disertori¹,

Merkl²,

Rolles³

2021

Preprint

Self Cite

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We study the non-linear supersymmetric hyperbolic sigma model H 2|2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2|2 model; its size is logarithmic in the size of the original model. 4 5

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A note on recurrence of the Vertex reinforced jump process and fractional moments localization

Collevecchio

Zeng

2018

Preprint

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We give a simple proof for recurrence of vertex reinforced jump process on Z d , under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on Z d is recurrent for strong reinforcement. Finally, we prove that the H (2|2) model on Z d localizes at strong disorder. Even though these results are well-known, we propose a unified approach, which also has the advantage to provide shorter proofs, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.

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Martingales and some generalizations arising from the supersymmetric hyperbolic sigma model

Cited by 3 publications

References 33 publications

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

A note on recurrence of the Vertex reinforced jump process and fractional moments localization

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