A Note on Large Deviations for 2D Coulomb Gas with Weakly Confining Potential

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.


Introduction and statement of the result
Given an infinite closed subset ∆ of C, consider the distribution of N particles x 1 , . . . , x N living on ∆ which interact like a Coulomb gas at inverse temperature β > 0 under an external potential. Namely, let P N be the probability distribution on ∆ N with density where the so-called potential V : ∆ → R is a continuous function which, provided ∆ is unbounded, grows sufficiently fast as |x| → ∞ so that For ∆ = R and β = 1 (resp. β = 2 and 4) such a density is known to match with the joint eigenvalue distribution of a N × N orthogonal (resp. unitary and unitary symplectic) invariant Hermitian random matrix [12]. A similar observation can be made when ∆ = C (resp. the unit circle T, the real half-line R + , the segment [0, 1]) by considering normal matrix models [5] (resp. the β-circular ensemble, the β-Laguerre ensemble, the β-Jacobi ensemble, see [8] for an overview). In this work, our interest lies in the limiting global distribution of the x i 's as N → ∞, that is the convergence of the empirical measure in the case where ∆ is unbounded and V satisfies a weaker growth assumption than usually presented in the literature, see (1.7). Note the µ N 's are random variables taking their values in the space M 1 (∆) of probability measures on ∆, that we equip with the usual weak topology. When ∆ = R, the almost sure convergence of (µ N ) N towards a non-random limit µ * V is classically known to hold under the hypothesis that there exists β ′ > 1 satisfying β ′ ≥ β such that lim inf |x|→∞ V (x) β ′ log |x| > 1, (1.4) that is, as |x| → ∞, the confinement effect due to the potential V is stronger than the repulsion between the x i 's. The limiting distribution µ * V is then characterized as the unique minimizer of the functional where we introduced the following variation of the weighted logarithmic kernel A stronger statement, first established by Ben Arous and Guionnet for a Gaussian potential V (x) = x 2 /2 [3] and later extended to arbitrary continuous potential V satisfying the growth condition (1.4) [1, Theorem 2.6.1] (see also [11,Theorem 5.4.3] for a similar statement with a slightly stronger growth assumption on V ), is that (µ N ) N satisfies a large deviation principle (LDP) on M 1 (∆) in the scale N 2 and good rate function . It is moreover known that µ * V has a compact support [1, Lemma 2.6.2]. A similar result is known to hold when ∆ = C, see e.g. [11,Theorem 5.4.9].
It is the aim of this work to show that such statements still hold, except that µ * V may not have compact support, when one allows the confining effect of the potential V to be of the same order of magnitude than the repulsion between the x i 's. Namely, we consider the following weaker growth condition: there exists β ′ > 1 satisfying We provide a statement when ∆ = R or C, and discuss later the case of more general ∆'s. More precisely, we will establish the following. (b) I V admits a unique minimizer µ * V on M 1 (∆).
A consequence of Theorem 1.1 (b) and (c), together with the Borel-Cantelli Lemma, is the almost sure convergence of (µ N ) N towards µ * V in the weak topology of M 1 (∆). Namely, if P stands for the probability measure induced by the product probability space N ∆ N , P N , we have where I N ∈ H N (C) is the identity matrix, dX the Lebesgue measure of H N (C) ≃ R N 2 and Z N a normalization constant. Such a matrix model is a variation of the Cauchy ensemble [8, Section 2.5]. Performing a spectral decomposition and integrating out the eigenvectors, it is known that the induced distribution for the eigenvalues is given by (1.1) with ∆ = R, β = 2, V (x) = log(1 + x 2 ), and some new normalization constant Z N . One can then compute, see Remark 2.2 below, that the minimizer of (1.5) is the Cauchy distribution where dx is the Lebesgue measure on R.
where dx stands for the Lebesgue measure on C ≃ R 2 , see Remark 2.2.

Remark 1.5. (Exponential tightness and compactification)
The proofs of the large deviation principles under the stronger growth assumption (1.4) presented in [3], [11], [1] follow a classical strategy in large deviation principles theory (see e.g [7] for an introduction), that is to control the deviations of (µ N ) N towards arbitrary small balls of M 1 (∆), and then prove an exponential tightness property for (µ N ) N : there exists a sequence of compact sets The exponential tightness is actually used to establish the large deviation upper bound, and plays no role in the proof of the lower one. Under the weaker growth assumption (1.7), it is not clear to the author how to prove the exponential tightness for (µ N ) N directly, and we thus prove Theorem 1.1 by using a different approach.
We adapt an idea of [9] and map C onto the Riemann sphere S, homeomorphic to the one-point compactification of C by the inverse stereographic projection T , then push-forward M 1 (C) to M 1 (S), and take advantage that the latter set is compact for its weak topology. More precisely, it will be seen that it is enough to establish upper bounds for the deviations of (T * µ N ) N , the push-forward of (µ N ) N by T , towards arbitrary small balls of M 1 (S). The latter fact is possible thanks to the explicit change of metric induced by T .
Our approach is still available for a large class of supports ∆ and for potentials V satisfying weaker regularity assumptions, justifying our choice to consider general ∆'s. Nevertheless, it is not the purpose of this note to establish in such a general setting the large deviation lower bound, which is a local property and in fact will be seen to be independent of the growth assumption for V . This is the reason why we restricted ∆ to be R or C in Theorem 1.1.
We first describe the announced compactification procedure in Section 2.1. Then, we study (T * µ N ) N and a related rate function in Section 2.2. From these informations, we are able to provide a proof for Theorem 1.1 in Section 2.3. Finally, we discuss in Section 3 some generalizations concerning the support of the Coulomb gas, the regularity of the potential and the compactification procedure of possible further interest.
2 Proof of Theorem 1.1 We first describe the compactification procedure. In this subsection, ∆ is an arbitrary unbounded closed subset of C.

Compactification
We consider the Riemann sphere, here parametrized as the sphere of R 3 centered in (0, 0, 1/2) of radius 1/2, , and T : C → S the associated inverse stereographic projection, namely the map defined by It is known that T an homeomorphism from C onto S \ {∞}, where ∞ = (0, 0, 1), so that (S, T ) is a one-point compactification of C. We write for convenience for the closure of T (∆) in S. For µ ∈ M 1 (∆), we denote by T * µ its push-forward by T , that is the measure on ∆ S characterized by for every Borel function f on ∆ S . Then the following Lemma holds.
Proof. T * is clearly continuous. The inverse of T * is given by push backward via T , that is, for any µ ∈ M 1 (∆ S ) satisfying µ({∞}) = 0, T * −1 µ(A) = µ(T (A)) for all Borel set A ⊂ ∆ S . To show the continuity of T * −1 , consider a sequence (µ N ) N in M 1 (∆ S ) with weak limit µ and assume that µ N ({∞}) = 0 for all N and µ({∞}) = 0. Then, for any ǫ > 0, the outer regularity of µ and the weak convergence of (µ N ) N towards µ yield the existence of a neighborhood B ⊂ ∆ S of ∞ such that lim sup which equivalently means that (T * −1 µ N ) N is tight. As a consequence, since f • T −1 is continuous on ∆ S for any continuous function f having compact support in ∆, the continuity of T * −1 follows.
The next step is to obtain an upper control on the deviation of (T * µ N ) N towards arbitrary small balls of M 1 (∆ S ).

Weak LDP upper bound for (T * µ N ) N
In this subsection, ∆ is an arbitrary unbounded closed subset of C, the potential V : ∆ → R ∪ {+∞} is a lower semi-continuous map satisfying the growth condition (1.7), and we assume there exists µ ∈ M 1 (∆) such that I V (µ) < +∞.
The change of metric induced by T is given by (see e.g. [2, Lemma 3.

4.2])
|T (x) − T (y)| = |x − y| where | · | stands for the Euclidean norm of R 3 (we identify C with {(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 = 0}). Note that by letting y → +∞ in (2.3), squaring and using the Pythagorean theorem, one obtains the useful relation From the potential V we then construct a potential V : ∆ S → R ∪ {+∞} in the following way. Set and Note that the growth assumption (1.7) is equivalent to V(∞) > −∞, so that V is lower semi-continuous on ∆ S . As a consequence the kernel is lower semi-continuous and bounded from below on ∆ S × ∆ S , and the functional is well-defined. One understands from (2.3), (2.5) and (2.2) that the potential V has been built so that the following relation holds Note that if ∆ = R (resp. ∆ = C) then ∆ S = S ∩ {(x 1 , x 2 , x 3 ) ∈ R 3 : x 2 = 0} is a circle (resp. ∆ C = S the full sphere). By rotational invariance, the minimizer of has to be the uniform measure U ∆ S of ∆ S , and thus the minimizer µ * V of I V is given by the push-backward T * −1 U ∆ S . Thus, if ∆ = R (resp. ∆ = C), an easy Jacobian computation involving polar (resp. spherical) coordinates yields that µ * V equals (1.8) (resp. (1.9)).
Given a metric d on M 1 (∆ S ), compatible with its weak topology (such as the Lévy-Prohorov metric, see [6]), we denote for the associated balls The following Proposition gathers all the informations concerning I V and (T * µ N ) N needed to establish Theorem 1.1 in the next Section.
(b) I V is strictly convex on the set where it is finite.

Proof. (a) It is equivalent to show that I V is lower semi-continuous. Since
when this integral makes sense, and note that if µ ∈ M 1 (∆ S ) then I(µ) ≥ 0. Since V is bounded from below and µ → V(z)dµ(z) is linear, it is enough to show that µ → I(µ) is strictly convex on the set where it is finite. Given µ, ν ∈ M 1 (∆ S ) having finite logarithmic energies, we have for any 0 < t < 1 We can easily compute the distribution for the z i 's induced by (1.1). Indeed, with V defined in (2.5)-(2.6), we obtain from the metric relations (2.3)-(2.4) that where λ stands for the push-forward by T of (the restriction of) the Lebesgue measure on ∆. As a consequence, we have (2.13) Then, with F V defined in (2.7), one can write (2.14) With F M V as in the proof of Proposition 2.3 (a) above, we have (2.15) Moreover, since P N -almost surely Note that by performing the change of variables z = T (x), using (2.4) and the growth assumption (1.7), it follows that and thus (2.17) yields 18) The continuity of the map provides by letting δ → 0 in (2.18)

19) and (c) is finally deduced by monotone convergence letting M → ∞ in (2.19).
Equipped with Proposition 2.3, we are now in position to prove Theorem 1.1 thanks to the compactification procedure described in Section 2.1.

Proof of Theorem 1.1
In this subsection, ∆ = R or C, and V : ∆ → R is a continuous map satisfying the growth assumption (1.7).
Proof of Theorem 1.1. (a) Since I V (µ) = +∞ for all µ ∈ M 1 (∆ S ) such that µ({∞}) > 0, we obtain from Lemma 2.1 and (2.9) that the levels sets of I V and I V are homeomorphic, namely for any α ∈ R Thus, Theorem 1.1 (a) follows from Proposition 2.3 (a).
(b) Theorem 1.1 (a) yields the existence of minimizers for I V on M 1 (∆). Since T * is a linear injection, it follows from (2.9) and Proposition 2.3 (b) that I V is strictly convex on the set where it is finite, which warrants the uniqueness of the minimizer.
(c),(d) It is enough to show that for any closed set F ⊂ M 1 (∆), and for any open set O ⊂ M 1 (∆), the latter quantity being finite. Let us first show (2.20). We have for any closed set F ⊂ M 1 (∆) that where clo(T * F) stands for the closure of T * F in M 1 (∆ S ). Inspired from the proof of [7, Theorem 4.1.11], we fix ǫ > 0, and introduce Then for any µ ∈ M 1 (∆ S ), Proposition 2.3 (c) provides the existence of δ µ > 0 such that lim sup Since M 1 (∆ S ) is compact, so is clo(T * F , and thus there exists a finite number of measures µ 1 , . . . , µ d ∈ clo(T * F such that As a consequence, it follows with (2.23) If ν ∈ clo(T * F), then either ν ∈ T * F or ν({∞}) > 0. Indeed, let (T * η N ) N be a sequence in T * F with limit ν satisfying ν({∞}) = 0. Lemma 2.1 yields η ∈ M 1 (∆) such that ν = T * η and moreover the convergence of (η N ) N towards η. Since F is closed, necessarily ν ∈ T * F. As a consequence, since I V (µ) = +∞ as soon as µ({∞}) > 0, we obtain from (2.9) inf µ∈ clo(T * F ) For any k large enough, define µ k ∈ M 1 (R) to be the normalized restriction of µ to the compact ∆ ∩ [−k, k] 2 . Then (µ k ) k converges towards µ as k → ∞ and one easily obtains from the monotone convergence theorem that

Generalizations
In this section we consider some generalizations of the result and the method presented in the previous sections.

Concerning the support of the Coulomb gas
A natural question is to ask if Theorem 1.1 still holds for more general supports ∆ and less regular potentials V , as suggested in the previous sections.
Let us emphasis that the compactification procedure presented in Section 2.1 and Proposition 2.3 hold under the only assumptions that ∆ is a closed subset of C and V : ∆ → R ∪ {+∞} is a lower semi-continuous map which satisfies the growth assumption (1.7), and such that there exists µ ∈ M 1 (∆) with I V (µ) < +∞. As a consequence, the proofs of Theorem 1.1(a), (b) and the upper bound (2.20) provided in Section 2.3 also hold under such a weakening of assumptions on V and ∆. A full large deviation principle would hold as soon as one can establish in this setting the lower bound (2.21) for µ N , or its equivalent for T * µ N , see Remark 2.4.

Concerning the compactification procedure
The main use of the compactification procedure was to avoid the use of exponential tightness to prove the large deviation upper bound. It turns out that the proof of (2.20) can be adapted without any substantial change to obtain a similar result in a more general setting that we present now.
Let X be a locally compact, but not compact, Polish space and consider a sequence (µ N ) N of random variables taking values in the space M 1 (X ) of Borel probability measures on X . Let ( X , T ) be a one-point compactification of X , that is a compact set X with an element ∞ ∈ X such that T : X → X is an homeomorphism on its image T (X) and X \ T (X ) = {∞}. Define T * to be the push-forward by T similarly as in (2.2). We equip M 1 ( X ) with its weak topology, so that it becomes a compact Polish space, and denotes B(µ, δ) the ball centered in µ ∈ M 1 ( X ) with radius δ > 0.  Then for any closed set F ⊂ M 1 (X ), Moreover, note that Φ has compact level sets (resp. is strictly convex on the set where it is finite) if and only if Φ • T * has (resp. is).
We mention that a similar strategy is used in [10] where a LDP is established for a two type particles Coulomb gas related to an additive perturbation of a Wishart random matrix model.