On Large Deviations for Gibbs Measures, Mean Energy and GammaConvergence
 8 Downloads
Abstract
We consider the random point processes on a measure space \((X,\mu _{0})\) defined by the Gibbs measures associated with a given sequence of Nparticle Hamiltonians \(H^{(N)}.\) Inspired by the method of Messer–Spohn for proving concentration properties for the laws of the corresponding empirical measures, we propose a number of hypotheses on \(H^{(N)}\) that are quite general but still strong enough to extend the approach of Messer–Spohn. The hypotheses are formulated in terms of the asymptotics of the corresponding mean energy functionals. We show that in many situations, the approach even yields a large deviation principle (LDP) for the corresponding laws. Connections to Gammaconvergence of (free) energy type functionals at different levels are also explored. The focus is on differences between positive and negative temperature situations, motivated by applications to complex geometry. The results yield, in particular, large deviation principles at positive as well as negative temperatures for quite general classes of singular mean field models with pair interactions, generalizing the 2D vortex model and Coulomb gases. In a companion paper, the results will be illustrated in the setting of Coulomb and Riesz type gases on a Riemannian manifold X, comparing with the complex geometric setting.
Keywords
Interacting particle systems Meanfield limit Large deviations Coulomb interactionMathematics Subject Classification
82C22 60F101 Introduction

Propose a number of quite general hypotheses on \(H^{(N)},\) formulated in terms of the corresponding mean energy functional \(E^{(N)}\)on \(\mathcal {P}(X^{N})\) (defined by formula 1.3) that are strong enough to extend the approach of Messer–Spohn.

Show that the approach also yields the stronger exponential concentration property in the sense of an LDP, almost “for free,” in several situations.

Explore some relations to the notion of Gammaconvergence of functionals: first by reformulating the approach of Messer–Spohn in terms of Gammaconvergence of the induced mean free energy functionals \(F_{\beta _{N}}^{(N)}\) on \(\mathcal {P}(\mathcal {P}(X))\) (formula (2.4)) and then by deducing a Gammaconvergence result for the sequence \(H^{(N)}/N\) on \(X^{N}.\)
Another motivation for the present paper comes from random matrix theory (or more generally Coulomb gases), which can be viewed as a vortex type model with \(\beta _{N}\sim N\) (and in particular \(\beta =\infty \)). The corresponding concentration property was established in [24], using the method of Messer–Spohn as in [15, 22]. Here we observe that, with a simple modification, the concentration property can be upgraded to an LDP (Corollary 1.2). In particular, this allows one to dispense with some technical assumptions (such as regularity properties of the corresponding pair interaction W(x, y) away from the diagonal) used in the different approaches to LDPs in [1, 2, 16, 34] (see Sect. 1.3).^{1}
Yet another motivation comes from approximation and sampling theory and, in particular, the problem of finding nearly minimal configurations for a given energy type interaction on a Riemannian manifold, in the spirit of [20, 32].
Let us also point out that that the restriction that X be compact can be removed if suitable growthassumptions of \(H^{(N)}\) at infinity are made, as in the settings in \(\mathbb {R}^{n}\) considered in [1, 2, 16, 18, 24, 34] (using appropriate tightness estimates). But in order to (hopefully) convey the conceptual simplicity of the arguments we stick with a compact X.
1.1 Hypotheses
 (H1) “Existence of a macroscopic mean energy \(E(\mu )''\): There exists a functional \(E(\mu )\) on \(\mathcal {P}(X)\) such that for any \(\mu \) in \(\mathcal {P}(X)\) satisfying \(E(\mu )<\infty \),Moreover, \(E(\mu _{0})<\infty \).$$\begin{aligned} \lim _{N\rightarrow \infty }E^{(N)}(\mu ^{\otimes N})=E(\mu ). \end{aligned}$$
 (H2) “Lower bound on the mean energy”: For any sequence of \(\mu ^{(N)}\) such that \(\Gamma _{N}:=(\delta _{N})_{*}\mu ^{(N)}\rightarrow \Gamma \) weakly in \(\mathcal {P}(Y)\), we have$$\begin{aligned} \liminf _{N\rightarrow \infty }E^{(N)}(\mu ^{(N)})\ge E(\Gamma ):=\int _{Y}E(\mu )\Gamma (\mu ). \end{aligned}$$

(H3) “Approximation property”: For any \(\mu \) such that \(E(\mu )<\infty \), there exists a sequence \(\mu _{j}\) converging weakly to \(\mu \) such that \(\mu _{j}\) has finite entropy with respect to \(\mu _{0}\) and satisfies \(E(\mu _{j})\rightarrow E(\mu ).\)
 (H4) “Mean energy/entropy compactness”: Ifwhere \(D^{(N)}(\mu ^{(N)})\) is the mean entropy, then the following convergence holds, after perhaps replacing \(\mu ^{(N)}\) by a subsequence such that \(\Gamma _{N}:=(\delta _{N})_{*}\mu ^{(N)}\rightarrow \Gamma \) weakly in \(\mathcal {P}(Y):\)$$\begin{aligned} E^{(N)}(\mu ^{(N)})\le C,\,\,\,D^{(N)}(\mu ^{(N)})\le C, \end{aligned}$$$$\begin{aligned} \lim _{N\rightarrow \infty }E^{(N)}(\mu ^{(N)})=\int _{Y}E(\mu )\Gamma (\mu ). \end{aligned}$$
Of course, the sign of the temperature may be switched by replacing \(H^{(N)}\) with \(H^{(N)},\) but the point is that, in practice, we will consider settings where the sign of \(H^{(N)}\) is fixed by the requirement that \(H^{(N)}\) be bounded from below (which essentially means that the system is assumed to be stable at zero temperature).
1.2 Large Deviation Results
We start with the simpler setting of positive temperature:
Theorem 1.1
Corollary 1.2
In the Euclidean setting and with \(M=2\), the previous corollary was established very recently in [18] using somewhat different methods (the results in [18] have also independently been generalized to the setting of the previous theorem and corollary in [19]).
We next turn to the case of negative temperature.
Theorem 1.3

For any \(\beta >\beta _{0},\) we have \(Z_{N,\beta }\le C_{\beta }^{N}\).
 For any \(\beta >\beta _{0},\) the measures \((\delta _{N})_{*}\left( e^{\beta H^{(N_{k})}}\mu _{0}^{\otimes N}\right) \) on \(\mathcal {P}(X)\) satisfy an LDP with speed N and rate functional$$\begin{aligned} \beta F_{\beta }(\mu )=\beta E(\mu )+D_{\mu _{0}}(\mu ). \end{aligned}$$
Corollary 1.4
The key observation in the proof of Corollary 1.4 is that the first point in Theorem 1.3 always implies, “for free,” a uniform estimate in the Orlitz (Zygmund) space \(L^{1}\hbox {Log}L^{1},\) so that some general Orlitz space duality results [26, 30] can be exploited in order to verify the hypothesis H4.
It seems natural to ask if the previous corollary can be generalized to the case when one only assumes the integrability condition that \(Z_{2,\beta }\) be finite when \(\beta >\beta _{0},\) for some (finite) negative number \(\beta _{0}.\) The following theorem gives an affirmative answer if one strengthens the integrability condition a bit:
Theorem 1.5
1.3 Relations to GammaConvergence at Different Levels
The proofs of the LDPs above are based on the Gammaconvergence of the corresponding free energy functionals \(F_{\beta _{N}}^{(N)}\) when viewed as functionals on the space \(\mathcal {P}(\mathcal {P}(X))\) (a similar approach is used in the dynamic setting considered in [10] where the assumptions H1 and H2 also appear naturally). Incidentally, as observed in the following corollary, the LDPs then imply the Gammaconvergence of the scaled Hamiltonians \(H^{(N)}/N\) when viewed as functionals on \(\mathcal {P}(X).\)
Corollary 1.6
Suppose that the Hamiltonians \(H^{(N)}\) satisfy H1 and H2. Then \(H^{(N)}/N\) Gammaconverges towards \(E(\mu )\) on \(\mathcal {P}(X).\) In particular, this applies to the finite order mean field Hamiltonians.
Relations between Gammaconvergence and large deviation principles have also been previously studied in [27] but from a somewhat different perspective (see also [13] for some related results).
1.4 Applications to the Coulomb Gas on a Riemannian Manifold
In the companion paper [6], the general large deviation results above are illustrated and further developed for Coulomb and Riesz type gases on a compact Ddimensional Riemannian manifold (X, g) and more generally for suitable compact subsets \(K\subset X\) (the case when \(\mu _{0}\) is a volume form and \(\beta >0\) has also indepedently been obtained in [19]).
Theorem 1.7

When \(\beta \in ]0,\infty [,\) the LDP holds if the measure \(\mu _{0}\) is nonpolar.

When \(\beta =\infty \), the LDP holds if \(\mu _{0}\) is nonpolar and \(\mu _{0}\) is determining for its support K.
 When \(D=2\) and \(\beta <\infty \), the LDP holds when \(\beta >4\pi d(\mu _{0}),\) where \(d(\mu _{0})\in [0,\infty [\) is the sup over all \(t>0\) such that there exists a positive constant C (depending on t) such thatas \(R\rightarrow 0,\) for any Riemannian ball \(B_{R}(x)\) of radius R centered at a given point x in X.$$\begin{aligned} \mu _{0}(B_{R}(x))\le CR^{t} \end{aligned}$$
More generally, an LDP as in the previous theorem is obtained in [6], when W(x, y) is taken as the integral kernel of the inverse of \((\Delta )^{p}\) and the (possible fractional) power p is in ]0, D / 2] (or even more generally: when \((\Delta )^{p}\) is replaced by a suitable pseudodifferential operator of order at most D). Then the last point in the previous theorem holds in the critical case \(p=D/2.\) However, the LDP for \(\beta =\infty \) appears to be rather subtle in the general setting and is only shown to hold when \(\mu _{0}\) is a volume form (or comparable to a volume form), except when \(p\le 2\), where it applies to measures \(\mu _{0}\) that are determining in a suitable sense.
Let us also point out that in the Euclidean setting of the Coulomb and Riesz gases in \(\mathbb {R}^{n},\) with \(\mu _{0}\) given by the Euclidean volume form and \(\beta _{N}\) of the order N, a refined “microscopic” large deviation principle “at the level of processes” is obtained in [25]. Such large deviation principles are beyond the scope of the present paper and seem to require different methods—the point here is rather to allow the measure \(\mu _{0}\) to be very singular (and the inverse temperature to be negative, in some cases).
2 Proofs of the Large Deviations Results
2.1 General Notation
We will denote by \(S_{N}\) the permutation group acting on \(X^{N}\) and by \(\mathcal {P}(X^{N})^{S_{N}}\) the space of symmetric measures \(\mu _{N}\) (i.e., \(S_{N}\)invariant) on \(X^{N}.\) Also note that, following standard practice, we will denote by C a generic constant whose value may change from line to line.
2.1.1 Entropy
2.2 Preliminaries
2.2.1 Large Deviation Principles
Let us start by recalling the general definition of a large deviation principle (LDP) for a sequence of measures.
Definition 2.1
 (i)
A function \(I:\mathcal {\,Y}\rightarrow ]\infty ,\infty ]\) is a rate function if it is lower semicontinuous. It is a good rate function if it is also proper; i.e., \(I^{1}]\infty ,a]\) is compact for any given \(a\in \mathbb {R}.\)
 (ii)
A sequence \(\Gamma _{N}\) of measures on Y satisfies a large deviation principle with speed \(r_{N}\) and rate function I if
Remark 2.2
The LDP is said to be weak if the upper bound is only assumed to hold when \(\mathcal {F}\) is compact. Anyway, we will only consider the case when Y is compact, and hence the notion of a weak LDP and an LDP then coincide (and moreover, any rate functional is automatically good).
Lemma 2.3
We also have the following simple lemma:
Lemma 2.4
The measures \(\tilde{\Gamma }_{N}:=(\delta _{N})_{*}(e^{\beta H^{(N)}}\mu _{0}^{\otimes N})\) are finite and satisfy the asymptotics in Bryc’s lemma with rate functional \(\tilde{I}(\mu )\) and speed N if and only if the corresponding probability measures \(\Gamma _{N}:=(\delta _{N})_{*}(\mu _{\beta }^{(N)})\) on \(\mathcal {P}(X)\) satisfy an LDP at speed N with rate functional \(I:=\tilde{I}C_{\beta },\) where \(C_{\beta }:=\inf _{\mathcal {\mu \in }\mathcal {P}(X)}\tilde{I}(\mu )\) and the sequence \(N^{1}\log Z_{N,\beta }\) is convergent in \(\mathbb {R}\) (and then the limit is equal to \(C_{\beta }).\)
Proof
2.2.2 GammaConvergence
Lemma 2.5
Assume that \(f_{j}\) Gammaconverges to f relative to \(\mathcal {S}\subset \mathcal {X}.\) Then \(f_{\mathcal {S}}\) is lower semicontinuous.
Proof
Consider a sequence \(s_{i}\rightarrow s\) in S. For each \(s_{i}\) we take a recovery sequence \(y_{i}^{(j)}\) in X converging to \(s_{i}.\) Setting \(y_{i}:=y_{i}^{(n_{i})},\) for a suitable increasing function \(i\mapsto n_{i},\) yields a sequence \(y_{i}\) in \(\mathcal {X}\) converging to s such that \(f(s_{i})\ge f_{n_{i}}(y_{i})1/i.\) Setting \(x_{n_{i}}:=y_{i}\) (and \(x_{j}:=s\) when j is not of the form \(j=n_{i}\) for any i) and using the first implication in the definition of the relative Gammaconvergence of the sequence \(f_{j},\) we thus deduce that \(\liminf _{i\rightarrow \infty }f(s_{i})\ge f(s),\) as desired. \(\square \)
Lemma 2.6
Proof
2.2.3 Legendre–Fenchel Transforms
2.3 Proof of Theorem 1.1
Now set \(Y:=\mathcal {P}(X).\) By embedding \(\mathcal {P}(X^{N}/S_{N})\) into \(\mathcal {P}(Y),\) using the pushward map \((\delta _{N})_{*},\) we can identify the mean free energies \(F_{\beta _{N}}^{(N)}\) with functionals on \(\mathcal {P}(Y),\) extended by \(\infty \) to all of \(\mathcal {P}(Y).\) We will identity Y with its image in \(\mathcal {P}(Y)\) under the embedding \(\mu \mapsto \delta _{\mu }.\)
The starting point of the proof of the LDP is the following reformulation of Bryc’s lemma in terms of the Legendre–Fenchel transform, using the Gibbs variational principle:
Lemma 2.7
Proof
Remark 2.8
Varadhan’s lemma implies that the converse of the previous lemma also holds.
In order to verify the criterion in the previous lemma, we will use the following lemma:
Lemma 2.9
Proof
First assume that \(\beta <\infty .\) The lower bound follows directly from hypotheses H1 and H2 together with the fact that the mean entropy functionals satisfy the lower bound in the Gammaconvergence (by subadditivity [31]; see also Theorem 5.5 in [21] for generalizations). To prove the existence of recovery sequences, we fix an element \(\Gamma \) of the form \(\delta _{\mu }\) and take the recovery sequence to be of the form \((\delta _{N})_{*}\mu ^{\otimes N}.\) Then the required convergence follows from H1 together with the product property (2.1.) Finally, when \(\beta =\infty \), the previous argument for the existence of a recovery sequence still applies as long as \(\mu \) satisfies \(D(\mu )<\infty .\) The general case then follows by a simple diagonal approximation argument using H3. \(\square \)
Now, since the limiting functional \(F_{\beta }(\Gamma )\) is affine and lower semicontinuous (by Lemma 2.5) and the set Y is extremal in \(\mathcal {P}(Y)\), the infimum of F on \(\mathcal {P}(Y)\) is attained in Y (for example, by Choquet’s theorem). Fixing a continuous function \(\Phi \) on \(C^{0}(Y)\) and replacing \(H^{(N)}\) with the new Hamiltonian \(H^{(N)}+N\delta _{N}^{*}(\Phi ),\) Lemma 2.6 thus shows that the criterion in Lemma 2.7 is satisfied. Hence the LDP holds with lower semicontinuous rate functional \(I(\mu )=f^{*}(\delta _{\mu }).\) Finally, extending I to \(\mathcal {P}(Y)\) by linearity, this means that \(I(\Gamma )\) is the Legendre–Fenchel transform of f, i.e., \(I=f^{*}.\) But in our case f is itself defined as \(f:=F^{*}\), and hence, \(I=F^{**}=F\) since F is convex (and even affine) and lower semicontinuous.
Remark 2.10
An inspection of the proof of Theorem 1.1 above reveals that, in the case \(\beta =\infty ,\) the hypothesis H3 may be replaced by the following weaker one:

(H3)’ The functional \(F_{\beta }\) Gammaconverges towards E, as \(\beta \rightarrow \infty \).
2.4 Proof of Corollary 1.2
2.5 Proof of Theorem 1.3
Remark 2.11
If one only wants to prove that the laws of \(\delta _{N}\) concentrate on the minima of \(F_{\beta }\) (rather than proving an LDP), it is enough to show that the convergence of the free energies hold for \(\Phi =0\) (as in the original approach in [28]). As revealed by the previous proof, this only requires that the hypothesis H4 holds for the particular sequence \(\mu _{\beta }^{(N)}.\)
2.6 Proof of Corollary 1.4
2.7 Proof of Theorem 1.5
Next we fix a continuous function \(\Phi \) on \(Y:=\mathcal {P}(X).\) Without loss of generality, we may as well assume that \(W,\Phi \ge 0.\)
Lemma 2.12
Proof
2.7.1 An Alternative Direct Proof of the Lower Semicontinuity of \(\beta F_{\beta }\)
A consequence of the LDP established above is that the corresponding (scaled) free energy functional \(\beta F_{\beta }\) is lsc on \(\mathcal {P}(X).\) As we show next, this could also be shown directly by establishing a macroscopic version of the hypothesis H4 (using Orlitz space duality). This indicates that there could be a more direct proof of the LDP that avoids Lemma 2.12, as discussed above.
Lemma 2.13
Under the assumptions of Theorem 1.5, the following holds: for any \(\beta >\beta _{0}\) the (scaled) free energy functional \(\beta F_{\beta }\) on \(\mathcal {P}(X)\) is lower semicontinuous.
Proof
2.8 Relations to GammaConvergence of \(E^{(N)}\) on \(\mathcal {P}(X):\) Proof of Corollary 1.6
3 Concluding Remarks
3.1 A Weaker form of the Hypothesis H2
 (H2’) For any sequence of \(x^{(N)}\in X^{N}\) such that \(\delta _{N}(x^{(N)})\rightarrow \mu \) weakly in \(\mathcal {P}(X)\), we have$$\begin{aligned} \liminf _{N\rightarrow \infty }\frac{1}{N}H^{(N)}(x^{(N)})\ge E(\mu ). \end{aligned}$$
Theorem 3.1
The conclusion of Theorem 1.1 remains valid if H2 is replaced by H2’.
Proof
A result essentially equivalent to the previous theorem appears in [13].
Combining Theorem 3.1 and Corollary 1.6 thus reveals that the hypotheses H1 and H2’ actually imply the Gammaconvergence of \(\frac{1}{N}H^{(N)}\) towards E on \(\mathcal {P}(X).\) But it seems unlikely that, in general, the assumption that \(\frac{1}{N}H^{(N)}\) Gammaconverges towards a functional E on \(\mathcal {P}(X)\) is sufficent to deduce an LDP (even if one also assumes H3). On the other hand, as shown in [5], one does get an LDP for any \(\beta \in ]0,\infty ]\) under an assumption of quasisuperharmonicity:
Theorem 3.2

The sequence \(\frac{1}{N}H^{(N)}\) on \(X^{N}\) (identified with a sequence of functions on \(\mathcal {P}(X)\)) Gammaconverges towards a functional E on \(\mathcal {P}(X)\).

\(H^{(N)}\) is uniformly quasisuperharmonic, i.e., \(\Delta _{x_{1}}H^{(N)}(x_{1},x_{2},\ldots x_{N})\le C\) on \(X^{N}\).
This is not hard to see when \(\beta =\infty ,\) but for \(\beta <\infty \), the proof hinges on a submean inequality for quasisubharmonic functions with a distortion factor that is subexponential in the dimension, proved in [5].
Footnotes
 1.
The Hamiltonians in the random matrix and Coulomb gas literature are usually scaled in a different way so that our zerotemperature \((\beta =\infty )\) corresponds to a fixed inverse temperature.
Notes
Acknowledgements
This work was supported by grants from the ERC and the KAW foundation. It is a pleasure to thank Sebastien Boucksom, Vincent Guedj, Philippe Eyssidieu, and Ahmed Zeriahi for the stimulating collaboration [9] and the referee for careful reading of the manuscript and many helpful remarks. Also thanks to the special issue editors Peter Forrester, Doug Hardin, and Sylvia Serfaty for the invitation to contribute to the special issue of Constructive Approximation.
References
 1.Ben Arous, G., Guionnet, A.: Large deviations for Wigner’s law and Voiculescu’s noncommutative entropy. Probab. Theory Rel. Fields 108(4), 517–542 (1997)MathSciNetCrossRefMATHGoogle Scholar
 2.Ben Arous, G., Zeitouni, O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)MathSciNetCrossRefMATHGoogle Scholar
 3.Berman, R.J.: Determinantal point processes and fermions on complex manifolds: large deviations and Bosonization. Comm. Math. Phys. 327(1), 1–47 (2014)MathSciNetCrossRefMATHGoogle Scholar
 4.Berman, R.J: Kähler–Einstein metrics, canonical random point processes and birational geometry. In: AMS Proceedings of the 2015 Summer Research Institute on Algebraic Geometry (to appear). arXiv:1307.3634
 5.Berman, R.J.: Large deviations for Gibbs measures with singular Hamiltonians and emergence of Kähler–Einstein metrics. Comm. Math. Phys. 354(3), 1133–1172 (2017)MathSciNetCrossRefMATHGoogle Scholar
 6.Berman, R.J.: Large Deviations for Gibbs Measures and Global Potential Theory: Riemannian Versus Kähler Manifolds (in preparation) Google Scholar
 7.Berman, R.J., Boucksom, S.: Growth of balls of holomorphic sections and energy at equilibrium. Invent. Math. 181(2), 337 (2010)MathSciNetCrossRefMATHGoogle Scholar
 8.Berman, R.J., Boucksom, S., Witt Nyström, D.: Fekete points and convergence towards equilibrium measures on complex manifolds. Acta Math. 207(1), 1–27 (2011)MathSciNetCrossRefMATHGoogle Scholar
 9.Berman, R.J., Boucksom, S., Eyssidieu, P., Guedj, V., Zeriahi, A.: Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties. Crelle’s J. (to appear). arXiv:1111.7158
 10.Berman, R.J., Onnheim, M.: Propagation of Chaos for a Class of First Order Models with Singular Mean Field Interactions. arXiv:1610.04327
 11.Bloom, T., Levenberg, N., Piazzon, P., Wielonsky, F: Bernstein–Markov: A Survey. Dolomites Res. Notes Approx. Vol. (Special Issue) 75–91 (2015). arXiv:1512.00739
 12.Bodineau, T., Guionnet, A.: About the stationary states of vortex systems. Ann. Inst. Henri Poincare Probab. Stat. 35, 205–237 (1999)MathSciNetCrossRefMATHGoogle Scholar
 13.Boucksom, S.: Limite thermodynamique et théorie du potentie. SMF Gazette Octobre. No. 146 (2015)Google Scholar
 14.Braides, A.: GammaConvergence for Beginners. Oxford University Press, Oxford (2002)CrossRefMATHGoogle Scholar
 15.Caglioti, E., Lions, P.L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for twodimensional Euler equations: a statistical mechanics description. Commun. Math. Phys. 143(3), 501–525 (1992)MathSciNetCrossRefMATHGoogle Scholar
 16.Chafaï, D., Gozlan, N., Zitt, P.A.: Firstorder global asymptotics for confined particles with singular pair repulsion. Ann. Appl. Probab. 24(6), 2371–2413 (2014)MathSciNetCrossRefMATHGoogle Scholar
 17.Dembo, A., Zeitouni O.: Large deviation techniques and applications. Corrected reprint of the second (1998) edition. In: Stochastic Modelling and Applied Probability, 38. Springer, Berlin (2010)Google Scholar
 18.Dupuis, P., Laschos, V., Ramanan, K.: Large Deviations for Empirical Measures Generated by Gibbs Measures with Singular Energy Functionals. arXiv:1511.06928
 19.García Zelada, D.: A Large Deviation Principle for Empirical Measures on Polish Spaces: Application to Singular Gibbs Measures on Manifolds. arXiv:1703.02680
 20.Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Not. Am. Math. Soc. 51(10), 1186–1194 (2004)MathSciNetMATHGoogle Scholar
 21.Hauray, M., Mischler, S.: On Kac’s chaos and related problems. J. Funct. Anal. (2014). arXiv:1205.4518
 22.Kiessling, M.K.H.: Statistical mechanics of classical particles with logarithmic interactions. Comm. Pure Appl. Math. 46, 27–56 (1993)MathSciNetCrossRefMATHGoogle Scholar
 23.Kiessling, M.K.H.: Statistical mechanics approach to some problems in conformal geometry. Phys. A: Stat. Mech. Appl. 279(1–4), 353–368 (2000)MathSciNetCrossRefGoogle Scholar
 24.Kiessling, Michael K.H., Spohn, H.: A note on the eigenvalue density of random matrices. Comm. Math. Phys. 199(3), 683–695 (1999)MathSciNetCrossRefMATHGoogle Scholar
 25.Leblé, T., Serfaty, S.: Large Deviation Principle for Empirical Fields of Log and Riesz Gases. arXiv:1502.02970
 26.Leonard, C.: Orlicz Spaces. http://leonard.perso.math.cnrs.fr/papers/LeonardOrlicz%20spaces.pdf
 27.Mariani, M.: A GammaConvergence Approach to Large Deviations. arXiv:1204.0640
 28.Messer, J., Spohn, H.: Statistical mechanics of the isothermal Lane–Emden equation. J. Stat. Phys. 29(3), 561–578 (1982)MathSciNetCrossRefGoogle Scholar
 29.Onsager: Statistical hydrodynamics. Supplemento al Nuovo Cimento 6:279–287 (1949)Google Scholar
 30.Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces, Volume 146 of Pure and Applied Mathematics. Marcel Dekker, New York (1991)Google Scholar
 31.Robinson, D.W., Ruelle, D.: Mean entropy of states in classical statistical mechanics. Comm. Math. Phys. 5, 288–300 (1967)MathSciNetCrossRefMATHGoogle Scholar
 32.Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell. 19(1), 5–11 (1997)MathSciNetCrossRefMATHGoogle Scholar
 33.Saff, E., Totik, V.: Logarithmic Potentials with Exteriour Fields. Springer, Berlin (1997) (with an appendix by Bloom, T)Google Scholar
 34.Serfaty, S.: Coulomb gases and Ginzburg–Landau vortices. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2015)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.