What do we actually know about logarithmic energy ? - MathOverflow

What do we actually know about logarithmic energy ?

2011-07-19T09:53:49Z

In potential theory, the $\textit{logarithmic energy}$ of a Radon measure $\mu$ acting on $\mathbb{C}$ is defined by $$I(\mu)=\iint\log\frac{1}{|x-y|}\mu(dx)\mu(dy).$$ Of course it is not well defined for all measures and may takes values in $[-\infty,+\infty]$. To avoid this annoying fact, one typically restrict to measures which integrate the logarithm around infinity, that is which satisfy the condition (C) $$ \int \log(1+|x|)\mu(dx)<+\infty,$$ so that $I(\mu)>-\infty$ thanks to $|x-y|\leq (1+|x|)(1+|y|) $. A fondamental fact is that if $\mu,\nu$ both satisfy (C), have finite logarithmic energy and $\mu(\mathbb{C})=\nu(\mathbb{C})$, then $I(\mu-\nu)\geq0$ and equality holds iff $\mu=\nu$ (we extend naturally the definition of $I$ to signed measures). I understand the condition (C) to be convenient, but maybe not sharp (one can imagine $\mu$ which does not satisfies (C) but with finite logarithmic energy). This leads to my first question :

What can we says about $I(\mu-\nu)$ when (at least one of) the measures do not satisfy (C) ?

This question is moreover motivated by the appearance of the logarithmic energies in random matrix theory (in large deviations rate functions) and in free probability (reinterpreted up to a sign as a non-commutative entropy by Voiculescu). In this setting, $I(\mu-\nu)$ is a natural candidate for a relative free entropy, and questions of geometric nature are bothering me : Let $A_{c}$ be the set of signed measures $\mu$ with finite logarithmic energy acting on $\mathbb{C}$ with total mass $\mu(\mathbb{C})=c$ such that if $\mu$ has Jordan decomposition $\mu^+-\mu^-$, then both $\mu^+$ and $\mu^-$ satisfy (C). Then the previous fact yields that $A_0$ is a pre-Hilbert space with scalar product $$ I(\mu,\nu)=\iint\log\frac{1}{|x-y|}\mu(dx)\nu(dy).$$ Note that $A_0$ is not complete since it is clearly not closed. Moreover, it is not hard to check that $A_0$ acts by translations on $A_1$ which inherits of a structure of affine space, leading to a metric on probability measures with finite logarithmic energy satisfying (C). It is now time for my second question :

What relations may exist between this metric and metrics compatible with the weak topology ? (e.g Prohorov's ? Levy / Bounded Lipshitz ? Wasserstein's ? ...) Or total variation norm ?

Answer by Margaret Friedland for What do we actually know about logarithmic energy ?

2011-07-19T16:07:26Z

Again, I do not have a ready answer to either of your questions, but I hope my pointers will somehow help.

1) There has been a lot of activity in weighted potential theory on complex manifolds, including the study of an energy functional which in the case $X = \mathbb{P}^1$, $\omega$=Fubini-Study form is (up to a factor) $E^*(\mu)=I(\mu-\omega)$. The relevant results are due to R. Berman, S. Boucksom, V. Guedj and A. Zeriahi. Since your background is more in analysis and probability than complex geometry, you may be better off checking the following papers:

Weighted Pluripotential Theory Results of Berman-Boucksom Authors: Norm Levenberg http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.4035v1.pdf

Pluripotential Energy Authors: Tom Bloom, Norm Levenberg http://arxiv.org/PS_cache/arxiv/pdf/1007/1007.2391v1.pdf

(and don't forget to talk to Zeriahi when you are in Toulouse!)

2) You will find some relation to weak convergence in: Voiculescu's entropy and potential theory Authors: Thomas Bloom http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.4551v1.pdf

(although none of the special metrics that you mention are discussed there).

3) If you want another related line of inquiry, a metric (and criteria of convergence) on compact, polynomially convex and L-regular subsets of $\mathbb{C}^n$ was proposed, using pluricomplex Green functions of these sets:

Klimek, Maciej Metrics associated with extremal plurisubharmonic functions. Proc. Amer. Math. Soc. 123 (1995), no. 9, 2763–2770.

Siciak, Józef On metrics associated with extremal plurisubharmonic functions. Bull. Polish Acad. Sci. Math. 45 (1997), no. 2, 151–161.

This seems like a particular case of your question 2, when the measures are Monge-Ampere measures. But the full answer would involve working out relations between convergence of measures and convergence of their supports.

Anyway, great questions!

Answer by Lutz Mattner for What do we actually know about logarithmic energy ?

2012-09-17T22:07:59Z

Concerning the first question: We have $I(\mu-\nu)>0$ whenever $I(\mu-\nu)$ is defined, finite or not, and $\mu$,$\nu$ are different signed Radon measures with equal total masses (or rather charges). This, with any finite dimensional Hilbert space in place of the plane, is Example 3.3 in http://www.ams.org/journals/tran/1997-349-08/S0002-9947-97-01966-1/home.html , where the assumption $\sigma\neq0$ is missing.