It is proved that a discrete group G is amenable if and only if for every unitary representation of G in an infinite-dimensional Hilbert space H the maximal uniform compactification of the unit sphere SH has a G-fixed point, that is, the pair (SH, G) has the concentration property in the sense of Milman. Consequently, the maximal U(H)equivariant compactification of the sphere in a Hilbert space...

— In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through BrunnMinkoswki inequalities), measure-theoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationships between these various viewpoints. Special emphasis is put on optimal mass transportation and the ...

In this work, we study concentration properties for vector valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in Rk. To this task, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions to earlier results by Pinelis in the Gaussian case, ...

Most applications of quantum information require many qubits, which means that they must be described using state spaces of very high dimension. The geometry of such spaces is invariably simple but often surprising. Subspaces, in particular, can be interpreted as quantum error correcting codes and, when the dimension is high enough, random subspaces form remarkably good codes. This is because i...

We prove what appears to be the first concentration of measure result for hidden Markov processes. Our bound is stated in terms of the contraction coefficients of the underlying Markov process, and strictly generalizes the Markov process concentration results of Marton (1996) and Samson (2000). Somewhat surprisingly, the hidden Markov process is at least as “concentrated” as its underlying Mark...

The aim of this paper is twofold. First, it shows the properties that regional economics is implicitly assuming when “relative” inequality measures, such as the Gini coefficient and the generalized entropy family of indexes, are used to quantify the geographic concentration of economic activity. Second, it proposes a new geographic concentration index that is based on an “absolute” inequality m...

Let pX,μq be a standard probability space. An automorphism T of this space has the weak Pinsker property if for every ε ą 0 it is isomorphic to a direct product of a Bernoulli shift and an automorphism of entropy less than ε. This property was introduced by Thouvenot, who conjectured that it holds for all measure-preserving systems. This paper proves Thouvenot’s conjecture. The proof applies wi...

The diameter of a graph is the maximum over vertices u and v of the distance between u and v. It is easy to show that graphs with high conductance must have low diameter. If the set of edges leaving a set must be large relative to the size of the set, then after taking neighborhoods a few times, one must encounter most of the edges. We now make this intuition precise. Recall from Lecture 7 the ...

Concentration of Measure refers to a techinque in probability for proving that certain random variables are unlikely to take values too much larger or smaller than their median. For example, consider the sum of d independent and uniformly chosen ±1 variables, x1, . . . , xd. The mean of the sum is obviously 0, and the symmetry of the distributions tells us that the median is 0 as well. The Cher...