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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 ?
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I think this shouldn't have the probability tag, since the only connection is that $\mu$ or $\mu^+$ might just happen to be a probability measure. –  Ricky Demer Jul 19 '11 at 10:04
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@Ricky: on the contrary, as the OP points out, $I$ plays an important role random matrix theory. –  Mark Meckes Jul 19 '11 at 13:13
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Very nice question, Adrien. –  Tom LaGatta Jul 19 '11 at 15:48
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Moreover, there is an intense activity about proving functional inequalities involving this logarithmic energy seen as a free probability equivalent of the famous Sanov's entropy, as free versions of the Poincaré inequality, transport inequality, log-sobolev [...]. See the works of Ledoux, Biane, Hiai, Petz etc –  Adrien Hardy Jul 19 '11 at 17:30
    
math.caltech.edu/SimonPapers/317.pdf explains the interest in logarithmic energy from another view point. Also an important aspect missing in the original post, is that one often wants to restrict the domain of definition of the measure. E.g. the real line, when considering ensembles of hermitian matrices... –  Helge Jul 19 '11 at 17:49

2 Answers 2

up vote 2 down vote accepted

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.

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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!

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Thank you a lot for the attention you give (gave) to my questions, I really appreciate ! But my questions really rely on "what can we say about logarithmic energy, or pluripotential energy, when we lose compactness". I have the impression that you sources deal with the compact case. Maybe you know the beautiful result of Simeonov "A weighted energy problem for a class of weakly admissible weights" (Houston Journal of Mathematics) which deal with log energy on compactification of $\mathbb{C}$ ? –  Adrien Hardy Jul 19 '11 at 17:39
    
My point is that I realized that many famous authors forget about the condition (C), and I don't really know what to think about that... I never spoke to Zeriahi, good idea ! Again, thank you for your help. –  Adrien Hardy Jul 19 '11 at 17:41
    
Bloom and Levenberg's papers are in the $\mathbb{C}^n$ setting. Basically, the results for compact case are applied with suitably adjusted weights. –  Margaret Friedland Jul 19 '11 at 18:18
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I did not know about Simeonov's result. I like the part about energy of a signed measure supported on the sphere. Perhaps I am spoiled after having worked on projective manifolds for a couple of years, but to me "weak admissibility" is really a continuity condition for a function on the projective space, and I would rather have the result stated and proved in this context. –  Margaret Friedland Jul 19 '11 at 18:50
    
Yes, I indeed had in mind the part about the energy on the sphere. Continuity ? I understand this condition of weak admissibility is a condition of "boundedness from below" on the sphere, can you develop ? –  Adrien Hardy Jul 21 '11 at 11:38

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