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Random Walks on Reductive Groups

Yves Benoist and Jean-François Quint
Publication Date: 
Number of Pages: 
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
[Reviewed by
Tushar Das
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Benoist and Quint have written an excellent text, one that will surely become a standard reference to introduce students to the fascinating nonabelian extension of the now-classical study of random walks. Instead of studying sums of independent identically distributed random real variables, the authors focus on products of independent identically distributed random matrices with real coefficients. The latter area is also studied under the name “random walks on linear groups” and was founded by the seminal work of Kesten, Furstenberg and Guivarc’h. Benoist and Quint’s text covers the study of such random walks on connected real semisimple algebraic groups. This class is wide enough to include the general/special linear, special orthogonal and the symplectic groups. With a view to arithmetic applications, the authors allow their matrix coefficients to lie in any local field (i.e. a finite extension of either the field of \(p\)-adic numbers, the field of Laurent series with coefficients in the integers mod \(p\) for \(p\) prime, or the reals). In spite of this generality, the book may be profitably read by assuming everywhere that the coefficients are real numbers. The neophyte reader may keep a concrete matrix group like \( \mathrm{SL}(2,\mathbb{R}) \) in mind when reading the text initially.

Classical answers to questions regarding the asymptotic behavior of such sequences often take one of two forms: those that describe the behavior in law (a.k.a. probability measure), versus those that describe the behavior of individual trajectories. The Central Limit Theorem (CLT), the Large Deviations Principle (LDP) and the Local Limit Theorem (LLT) are statements in law, whereas the Law of Large Numbers (LLN) and the Law of the Iterated Logarithm (LIL) are statements about almost every trajectory. The tools employed in proving these statements come from a variety of areas: for instance the Doob Martingale Theorem from probability theory, the Birkhoff Ergodic Theorem from ergodic theory, and the Fourier Inversion Theorem from classical harmonic analysis. Certain new ideas/perspectives are also necessary: the LLN will follow from the ergodic theory of trajectories of an associated Markov chain on the projective space or flag variety, while spectral data from a complexified transfer (Ruelle-Perron-Frobenius) operator can be used in combination with the Fourier Inversion Theorem to prove the CLT, LIL, LDP and LLT.

Though the book provides readers with a wonderful work of synthesis, it also serves as a background reference for those interested in accessing the authors’ celebrated suite of papers:

  • “Mesures stationnaires et fermés invariants des espaces homog\`enes” (Ann. of Math. (2) 174 (2011), no. 2, 1111--1162)
  • “Random walks on finite volume homogeneous spaces” (Invent. Math. 187 (2012), no. 1, 37--59)
  • “Stationary measures and invariant subsets of homogeneous spaces (II)” (J. Amer. Math. Soc. 26 (2013), no. 3, 659--734)
  • “Stationary measures and invariant subsets of homogeneous spaces (III)” (Ann. of Math. (2) 178 (2013), no. 3, 1017--1059)

An embellished version of the 10th Takagi Lectures delivered by Benoist in 2012 was published as “Introduction to random walks on homogeneous spaces”, by Benoist-Quint (Jpn. J. Math. 7 (2012), no. 2, 135--166). The reader interested in moving beyond the book being reviewed (towards Benoist-Quint’s suite of applications) could start with the Takagi lectures, and then move to the preprint versions of the papers mentioned above (and yet more!) at Benoist’s research page. There is also a recent English translation of the first paper in their series, which originally appeared in French.

I congratulate the authors on their well-written and timely offering, and strongly recommend that libraries order a copy of this excellent text!

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.

See the table of contents in the publisher's webpage.