Impurity models and products of random matrices

Alain Comtet; Yves Tourigny

doi:10.1093/oso/9780198797319.001.0001

Impurity models and products of random matrices

Alain Comtet, Yves Tourigny

Research output: Chapter in Book/Report/Conference proceeding › Chapter in a book

358 Downloads (Pure)

Abstract

This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools.
After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.

Original language	English
Title of host publication	Stochastic Processes and Random Matrices
Subtitle of host publication	Lecture Notes of the Les Houches Summer School 2015
Editors	Gregory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O'Connell, Leticia F. Cugliandolo
Place of Publication	Oxford
Publisher	Oxford University Press
Chapter	11
Number of pages	71
Volume	104
ISBN (Print)	9780198797319
DOIs	https://doi.org/10.1093/oso/9780198797319.001.0001
Publication status	Published - 10 Aug 2017

Publication series

Name	Lecture Notes of the Les Houches Summer School
Publisher	OUP

Access to Document

10.1093/oso/9780198797319.001.0001

Full-text PDF (accepted author manuscript)
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via OUP at https://global.oup.com/academic/product/stochastic-processes-and-random-matrices-9780198797319?cc=gb&lang=en&. Please refer to any applicable terms of use of the publisher.
Accepted author manuscript, 614 KB

Handle.net

Persistent link

Dr Yves J M Tourigny
- Y.Tourignybristol.acuk
- Probability, Analysis and Dynamics
- School of Mathematics - Senior Lecturer in Numerical Analysis
- Applied Mathematics
- Mathematical Physics
Person: Academic , Member

Cite this

Comtet, A., & Tourigny, Y. (2017). Impurity models and products of random matrices. In G. Schehr, A. Altland, Y. V. Fyodorov, N. O'Connell, & L. F. Cugliandolo (Eds.), Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School 2015 (Vol. 104). (Lecture Notes of the Les Houches Summer School). Oxford University Press. https://doi.org/10.1093/oso/9780198797319.001.0001

Comtet, Alain ; Tourigny, Yves. / Impurity models and products of random matrices. Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School 2015. editor / Gregory Schehr ; Alexander Altland ; Yan V. Fyodorov ; Neil O'Connell ; Leticia F. Cugliandolo. Vol. 104 Oxford : Oxford University Press, 2017. (Lecture Notes of the Les Houches Summer School).

@inbook{d8b3609b9c75459288af14a0253a70f0,

title = "Impurity models and products of random matrices",

abstract = "This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools.After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.",

author = "Alain Comtet and Yves Tourigny",

year = "2017",

month = aug,

day = "10",

doi = "10.1093/oso/9780198797319.001.0001",

language = "English",

isbn = "9780198797319",

volume = "104",

series = "Lecture Notes of the Les Houches Summer School",

publisher = "Oxford University Press",

editor = "Gregory Schehr and Alexander Altland and Fyodorov, {Yan V.} and Neil O'Connell and Cugliandolo, {Leticia F.}",

booktitle = "Stochastic Processes and Random Matrices",

address = "United Kingdom",

}

Comtet, A & Tourigny, Y 2017, Impurity models and products of random matrices. in G Schehr, A Altland, YV Fyodorov, N O'Connell & LF Cugliandolo (eds), Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School 2015. vol. 104, Lecture Notes of the Les Houches Summer School, Oxford University Press, Oxford. https://doi.org/10.1093/oso/9780198797319.001.0001

Impurity models and products of random matrices. / Comtet, Alain; Tourigny, Yves.
Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School 2015. ed. / Gregory Schehr; Alexander Altland; Yan V. Fyodorov; Neil O'Connell; Leticia F. Cugliandolo. Vol. 104 Oxford: Oxford University Press, 2017. (Lecture Notes of the Les Houches Summer School).

Research output: Chapter in Book/Report/Conference proceeding › Chapter in a book

TY - CHAP

T1 - Impurity models and products of random matrices

AU - Comtet, Alain

AU - Tourigny, Yves

PY - 2017/8/10

Y1 - 2017/8/10

N2 - This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools.After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.

AB - This is an introduction to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2 by 2 case. The notion of impurity model--- that is, a system in which the interactions are highly localised--- links the two themes and enables their study by elementary mathematical tools.After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.

U2 - 10.1093/oso/9780198797319.001.0001

DO - 10.1093/oso/9780198797319.001.0001

M3 - Chapter in a book

SN - 9780198797319

VL - 104

T3 - Lecture Notes of the Les Houches Summer School

BT - Stochastic Processes and Random Matrices

A2 - Schehr, Gregory

A2 - Altland, Alexander

A2 - Fyodorov, Yan V.

A2 - O'Connell, Neil

A2 - Cugliandolo, Leticia F.

PB - Oxford University Press

CY - Oxford

ER -

Comtet A, Tourigny Y. Impurity models and products of random matrices. In Schehr G, Altland A, Fyodorov YV, O'Connell N, Cugliandolo LF, editors, Stochastic Processes and Random Matrices: Lecture Notes of the Les Houches Summer School 2015. Vol. 104. Oxford: Oxford University Press. 2017. (Lecture Notes of the Les Houches Summer School). doi: 10.1093/oso/9780198797319.001.0001

Impurity models and products of random matrices

Abstract

Publication series

Access to Document

Handle.net

Fingerprint

Profiles

Dr Yves J M Tourigny

Cite this