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In these notes we review the S-matrix theory in (1+1)-dimensional integrable models, focusing mainly on the relativistic case. Once the main definitions and physical properties are introduced, we discuss the factorization of scattering processes due to integrability. We then focus on the analytic properties of the two-particle scattering amplitude and illustrate the derivation of the S-matrices for all the possible bound states using the so-called bootstrap principle. General algebraic structures underlying the S-matrix theory and its relation with the form factors axioms are briefly mentioned. Finally, we discuss the S-matrices of sine-Gordon and SU(2), SU(3) chiral Gross-Neveu models. This is part of a collection of lecture notes for the Young Researchers Integrability School, organized by the GATIS network at Durham University on 6-10 July 2015.

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds. We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors.

Book Source: Digital Library of India Item 2015.163813 dc.contributor.author: J H M Wedderburn dc.contributor.other: Indian Institute Of Science dc.date.accessioned: 2015-07-06T23:10:52Z dc.date.available: 2015-07-06T23:10:52Z dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1934 dc.identifier.barcode: 1010010031990 dc.identifier.origpath: /data/upload/0031/995 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/163813 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 210 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Vishwas dc.publisher: American Mathematical Society Newyork dc.rights: Out_of_copyright dc.source.library: Iisc Library dc.subject.classification: Mathematics dc.title: Lectures On Matrices

Book Source: Digital Library of India Item 2015.163813 dc.contributor.author: J H M Wedderburn dc.contributor.other: Indian Institute Of Science dc.date.accessioned: 2015-07-06T23:10:52Z dc.date.available: 2015-07-06T23:10:52Z dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1934 dc.identifier.barcode: 1010010031990 dc.identifier.origpath: /data/upload/0031/995 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/163813 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 210 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Vishwas dc.publisher: American Mathematical Society Newyork dc.rights: Out_of_copyright dc.source.library: Iisc Library dc.subject.classification: Mathematics dc.title: Lectures On Matrices

Book Source: Digital Library of India Item 2015.163813 dc.contributor.author: J H M Wedderburn dc.contributor.other: Indian Institute Of Science dc.date.accessioned: 2015-07-06T23:10:52Z dc.date.available: 2015-07-06T23:10:52Z dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1934 dc.identifier.barcode: 1010010031990 dc.identifier.origpath: /data/upload/0031/995 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/163813 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 210 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Vishwas dc.publisher: American Mathematical Society Newyork dc.rights: Out_of_copyright dc.source.library: Iisc Library dc.subject.classification: Mathematics dc.title: Lectures On Matrices

Book Source: Digital Library of India Item 2015.163813 dc.contributor.author: J H M Wedderburn dc.contributor.other: Indian Institute Of Science dc.date.accessioned: 2015-07-06T23:10:52Z dc.date.available: 2015-07-06T23:10:52Z dc.date.digitalpublicationdate: 0000-00-00 dc.date.citation: 1934 dc.identifier.barcode: 1010010031990 dc.identifier.origpath: /data/upload/0031/995 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/163813 dc.description.scanningcentre: IISc, Bangalore dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 210 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Vishwas dc.publisher: American Mathematical Society Newyork dc.rights: Out_of_copyright dc.source.library: Iisc Library dc.subject.classification: Mathematics dc.title: Lectures On Matrices