A Random Matrix Model From Two Dimensional Yang-Mills Theory

Xu, Feng

doi:10.1007/s002200050242

Cited by 44 publications

(48 citation statements)

References 5 publications

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“…Here is an example of results of this type (related results are proved in [36]). Theorem 6.1 Let k be a positive integer, {T r,n } k r=1 be a set of n × nmatrices, such that sup r≤k; k,l,n∈N n −1 Tr(T * r,n T r,n ) l < ∞, (6.4) and let U n be the unitary and Haar-distributed random matrix.…”

Section: Discussionmentioning

confidence: 81%

On the Law of Addition of Random Matrices

Pastur¹,

Vasilchuk

2000

Communications in Mathematical Physics

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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U * n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.

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Section: Discussionmentioning

confidence: 81%

On the Law of Addition of Random Matrices

Pastur¹,

Vasilchuk

2000

Communications in Mathematical Physics

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“…The proof of Theorem 5.4.10 we presented follows the characterization of the law of free unitary variables by a Schwinger-Dyson equation given in [Voi99, Proposition 5.17] and the ideas of [CoMG06]. Other proofs were given in terms of Weingarten functions in [Col03] and with a more combinatorial approach in [Xu97]. For uses of master loop (or Schwinger-Dyson) equations in the physics literature, see e.g.…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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“…It would be of interest to relate this limit to the approach described by Singer [12]. It should also be possible to connect the freenoise process to the large-N limit of Wilson loop expectations described in the physics literature as well as the mathematical works of Biane [1], Lévy [9], and Xu [17]. Results concerning the large-N limit of Wilson loop expectations are reviewed and developed in [11].…”

Section: Discussionmentioning

confidence: 99%

“…The large-N limit of (13) has been studied by Biane [1], Xu [17], the author [11], and, more extensively, by Lévy [9].…”

Section: Stochastic Holonomymentioning

confidence: 99%