Mark Meckes's research

Magnitude and diversity

Magnitude and Holmes–Thompson intrinsic volumes of convex bodies.
Canadian Mathematical Bulletin 66 (2023) no. 3, pp. 854–867.
arXiv / published version
Spaces of extremal magnitude (with T. Leinster).
Proc. Amer. Math. Soc., 151 (2023), pp. 3967–3973.
arXiv / published version
On the magnitude and intrinsic volumes of a convex body in Euclidean space.
Mathematika 66 (2020) no. 2, 343–355.
arXiv / published version
The magnitude of a metric space: from category theory to geometric measure theory (with T. Leinster).
Measure Theory in Non-Smooth Spaces, 156–193, DeGruyter Open, 2017.
arXiv / published version
Maximizing diversity in biology and beyond (with T. Leinster).
Entropy 18 (2016) no. 3, article 88.
arXiv / published version
Note: Due to the journal's editorial policy, the numbering of theorems, examples, etc. in this paper was changed between the arXiv preprint and the published version.
Magnitude, diversity, capacities, and dimensions of metric spaces.
Potential Anal. 42 (2015) no. 2, 549–572.
arXiv / published version
Positive definite metric spaces.
Positivity 17 (2013) no. 3, 733–757.
arXiv / published version
Note: The last paragraph of the published version of misstates the consequences for magnitude dimension of Theorems 4.4 and 4.5. The discussion around those results has been revised in the arXiv version to clarify this matter.

Random matrix theory

Fluctuations of the spectrum in rotationally invariant random matrix ensembles (with E. Meckes).
Random Matrices Theory Appl. 10 (2021) no. 3, Article No. 2150025.
arXiv / published version
Random matrices with prescribed eigenvalues and expectation values for random quantum states (with E. Meckes).
Trans. Amer. Math. Soc. 373 (2020) no. 7, 5141–5170.
arXiv / published version
A sharp rate of convergence for the empirical spectral measure of a random unitary matrix (with E. Meckes).
Zapiski Nauchnykh Seminarov POMI 457 (2017), 276–285 / Journal of Mathematical Sciences.
arXiv / published version
Rates of convergence for empirical spectral measures: a soft approach (with E. Meckes).
Convexity and Concentration, 157–181, IMA Volumes in Mathematics and its Applications 161, Springer, 2017.
arXiv / published version
Self-similarity in the circular unitary ensemble (with E. Meckes).
Discrete Analysis, 2016:9, 15 pp.
arXiv / journal's editorial introduction (published version is on the arXiv)
A rate of convergence for the circular law for the complex Ginibre ensemble (with E. Meckes).
Ann. Fac. Sci. Toulouse Math. Series 6, 24 (2015) no. 1, 93–117.
arXiv / published version
Spectral measures of powers of random matrices (with E. Meckes).
Electron. Commun. Probab. 18 (2013) no. 78, 1–13.
arXiv / published version
Concentration and convergence rates for spectral measures of random matrices (with E. Meckes).
Probab. Theory Related Fields 156 (2013), 145–164.
arXiv / published version
The spectra of random abelian G-circulant matrices.
ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) no. 2, 435–450.
arXiv / published version
Concentration for noncommutative polynomials in random matrices (with S. Szarek).
Proc. Amer. Math. Soc. 140 (2012), 1803–1813.
arXiv / published version
Another observation about operator compressions (with E. Meckes).
Proc. Amer. Math. Soc. 139 (2011), 1433–1439.
arXiv / published version
Some results on random circulant matrices.
High Dimensional Probability V: The Luminy Volume, 213–223, IMS Collections 5, Institute of Mathematical Statistics, Beachwood, OH, 2009.
arXiv / published version
On the spectral norm of a random Toeplitz matrix.
Electron. Commun. Probab. 12 (2007), 315–325.
arXiv / published version
Concentration of norms and eigenvalues of random matrices.
J. Funct. Anal. 211 (2004) no. 2, 508–524.
arXiv / published version

Probability in convexity and convexity in probability

On the equivalence of modes of convergence for log-concave measures (with E. Meckes).
Geometric Aspects of Functional Analysis, 385–394, Lecture Notes in Math. 2116, Springer, 2014.
arXiv / published version
Gaussian marginals of convex bodies with symmetries.
Beiträge Algebra Geom. 50 (2009) no. 1, 101–118.
arXiv / published version
The central limit problem for random vectors with symmetries (with E. Meckes).
J. Theoret. Probab. 20 (2007), 697–720.
arXiv / published version
Note: The arXiv preprint contains a section on background on Stein's method which does not appear in the published version. As a result, some theorem numbers are different in the two versions.
Some remarks on transportation cost and related inequalities.
Geometric Aspects of Functional Analysis, 237–244, Lecture Notes in Math. 1910, Springer, 2007.
arXiv / published version
Sylvester's problem for symmetric convex bodies and related problems.
Monatsh. Math. 145 (2005) no. 4, 307–319.
arXiv / published version
Note: The published version contains several references to the literature which are missing in the arXiv preprint, and has improved proofs of Propositions 13 and 16.
Volumes of symmetric random polytopes.
Arch. Math. 82 (2004) no. 1, 85–96.
arXiv / published version

Other

Quenched central limit theorem in a corner growth setting (with H.C. Gromoll and L. Petrov).
Electron. Commun. Probab. 23 (2018), paper no. 101, 1–12.
arXiv / published version

Random Matrices from the Classical Compact Groups: a Panorama

While visiting Oxford University in 2020—21 I gave a series of expository talks on random matrices from the classical compact groups. Slides and videos are available here.

Back to Mark Meckes's home page.

Mark Meckes's research

Papers by topic

Magnitude and diversity

Random matrix theory

Probability in convexity and convexity in probability

Other

Magnitude and diversity

Random matrix theory

Probability in convexity and convexity in probability

Other

Random Matrices from the Classical Compact Groups: a Panorama