Mark Meckes's papers

Random matrices

• Some results on random circulant matrices. Preprint
  To appear in the proceedings of High Dimensional Probability V.
  Abstract: This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.

• On the spectral norm of a random Toeplitz matrix. Published version / Preprint
  Electron. Commun. Probab. 12 (2007), 315-325.
  Abstract: Suppose that T_n is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of T_n is of the order √(n log n). The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been studied in the literature.

• Concentration of norms and eigenvalues of random matrices. Published version / Preprint
  J. Funct. Anal. 211 (2004) no. 2, 508-524.
  Abstract: We prove concentration results for l_pⁿ operator norms of rectangular random matrices and eigenvalues of self-adjoint random matrices. The random matrices we consider have bounded entries which are independent, up to a possible self-adjointness constraint. Our results are based on an isoperimetric inequality for product spaces due to Talagrand.

Asymptotic convex geometry

• Gaussian marginals of convex bodies with symmetries. Published version / Preprint
  Beiträge Algebra Geom. 50 (2009) no. 1, pp. 101-118.
  Abstract: We prove Gaussian approximation theorems for specific k-dimensional marginals of convex bodies which possess certain symmetries. In particular, we treat bodies which possess a 1-unconditional basis, as well as simplices. Our results extend recent results for 1-dimensional marginals due to E. Meckes and the author.

• The central limit problem for random vectors with symmetries (with E. Meckes). Published version / Preprint
  J. Theoret. Probab. 20 (2007), 697-720.
  Note: the preprint contains a section on background on Stein's method which does not appear in the published version.
  Abstract: Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein's method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry and we give a brief introduction to the classical method. The spherically symmetric case is treated by a variation of Stein's method which is adapted for continuous symmetries.

• Some remarks on transportation cost and related inequalities.
  Published version / Preprint
  Geometric Aspects of Functional Analysis, 237-244, Lecture Notes in Math. 1910, Springer, Berlin, 2007.
  Abstract: We discuss transportation cost inequalities for uniform measures on convex bodies, and connections with other geometric and functional inequalities. In particular, we show how transportation inequalities can be applied to the slicing problem, and give a new log-Sobolev-type inequality for bounded domains in Rⁿ.

Geometric probability

• Sylvester's problem for symmetric convex bodies and related problems. Published version / Preprint
  Monatsh. Math. 145 (2005) no. 4, 307-319.
  Note: The published version contains several references to the literature which are missing in the preprint, and has improved proofs of Propositions 13 and 16.
  Abstract: We consider moments of the normalized volume of a symmetric or nonsymmetric random polytope in a fixed symmetric convex body. We investigate for which bodies these moments are extremized, and calculate exact values in some of the extreme cases. We show that these moments are maximized among planar convex bodies by parallelograms.

• Volumes of symmetric random polytopes. Published version / Preprint
  Arch. Math. 82 (2004) no. 1, 85-96.
  Abstract: We consider the moments of the volume of the symmetric convex hull of independent random points in an n-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for n points when the given body is B_qⁿ (and all of the moments for the case q = 2), and derive from these the asymptotic behavior of the expected volume of a random simplex in those bodies.