A localization theorem for the planar Coulomb gas in an external field

We examine a Coulomb gas consisting of $n$ identical repelling point charges at an arbitrary inverse temperature $\beta$, subjected to a suitable external field. We prove that the gas is effectively localized to a small neighbourhood of the 'droplet' -- the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to $$\sqrt{\dfrac {\log n}{\beta n}}.$$ This order of magnitude is known to be 'tight' when $\beta=1$ and the external field is radially symmetric.


Introduction and Main Result
To get started, we briefly recapitulate some notions pertinent to the Coulomb gas with respect to an external field in the plane C = R 2 .
The planar Coulomb gas is a random configuration consisting of many (but finitely many) identical repelling point charges {ζ i } n 1 in C. To keep the system from dispersing to infinity we assume the presence of an external field nQ where Q is a suitable extended real-valued function defined on C, large near infinity in the sense that (1.1) lim inf ζ→∞ Q(ζ) 2 log |ζ| > 1.
The function Q, which is called an external potential, is fairly general but not quite arbitrary; precise assumptions are given below.
To a plane configuration {ζ j } n 1 we associate the Hamiltonian (or energy) Here and throughout we write dA for the Lebesgue measure in C divided by π and dA n (ζ 1 , . . . , ζ n ) = dA(ζ 1 ) · · · dA(ζ n ) is the corresponding product measure; the constant Z We shall now gradually become explicit about our precise assumptions, introducing simultaneously our basic objects of study.
We will always assume that the potential function Q : C → R ∪ {+∞} is lower semi-continuous and that the set Σ := {Q < +∞} has non-empty interior Int Σ. We will also suppose that Q is C 2 -smooth in Int Σ.
Given a potential of this type and a finite, compactly supported Borel measure µ on C, we define its logarithmic Q-energy by (Here 'µ(Q)' is shorthand for C Q dµ.) It is shown in [17] that there is a unique equilibrium measure σ of total mass 1, which minimizes I Q [µ] over all compactly supported Borel probability measures µ. Its support, which we denote by the symbol S = S[Q] := supp σ is called the droplet in external potential Q.
It is well known that the equilibrium measure is absolutely continuous and takes the form (1.5) dσ = ∆Q · 1 S dA.
Here and in what follows we normalize the Laplacian by Observe that since σ is a positive measure, Q is necessarily subharmonic on the support S.
We now list additional conditions which we always assume to be satisfied in the sequel, except when otherwise is explicitly stated.
(ii) The potential Q is strictly subharmonic in a neighbourhood of S.
(iii) The boundary ∂S has finitely many components.
(iv) Each component of ∂S is an everywhere C 1 -smooth Jordan curve.
(v) S * = S where S * is the coincidence set for the obstacle problem, given in Section 2.
Some of these conditions are assumed merely for convenience and may be relaxed, cf. subsections 4.1-4.2.
For a point ζ in the complement S c := C \ S we will denote its Euclidean distance to S by the symbol Associated to a random sample {ζ j } n 1 we define the number (1.6) We regard D n as a random variable with respect to the Gibbs measure, which thus represents the distance from the droplet to the vacuum.
In what follows, we tacitly assume that n is 'large enough', i.e., n ≥ n 0 for some suitable n 0 .
We are now ready to state our first main result.
Theorem 1. Let β = β n be a possibly n-dependent inverse temperature obeying βn → ∞ as n → ∞. Then there exists a sequence µ = µ n of positive numbers with and constants c > 0 and a > 0, such that for each real t satisfying t ≤ aβn we have the estimate Here c and a depend only on Q; c can be taken as any positive number with c < c 0 where (1.9) c 0 := min{∆Q(η); η ∈ ∂S}.
Remark. The meaning of the notation a n b n is that a n ≤ Cb n for all large n, where C is some unspecified positive constant. The symbol a n ≍ b n means a n b n and b n a n .
Recall from [12] that (for any fixed β > 0) the system {ζ j } n 1 tends to follow the equilibrium measure in the sense that for each bounded continuous function f . The convergence in (1.10) implies, in a loose sense, that that the particles are likely to stay in the immediate vicinity of the droplet. It could be said that Theorem 1 gives more detailed information about exactly how 'localized' the gas is about the droplet.
To illuminate this localization, we may observe that if we fix a β > 0 and choose t = t n so that t n → ∞ and t n / log n → 0 as n → ∞, then (1.8) implies Hence if A satisfies the premise in (1.11), then the gas is effectively localized to the set of ζ with The estimate (1.12) might be compared with earlier results on the distribution of the spectral radius of certain types of normal random matrices, due to Rider [16] for the Ginibre ensemble and Chafaï and Péché [8] for more general ensembles corresponding to radially symmetric potentials Q.
To appreciate this connection, we recall that we can interpret the Coulomb gas {ζ j } n j=1 in external potential Q at inverse temperature β = 1 as eigenvalues of normal random matrices. (See [9,11] for details.) Indeed, let us temporarily assume that Q is radially symmetric and that the droplet is a disc centered at 0 of radius R. A normal matrix with eigenvalues {ζ j } n 1 has its spectral radius equal to max 1≤j≤n |ζ j |, which is in the present case essentially equal to R + D n where D n is the random variable (1.6). Using this observation, we shall show in Subsection 4.3 that the estimate (1.12) comes close to earlier results due to Rider [16] and Chafaï-Péché [8], in the sense that the order of magnitude of our obtained localization is comparable with what is obtained in the indicated papers.
Plan of this paper. Sections 2 and 3 are devoted to our proof of Theorem 1. In Section 4 we will state and prove two generalized versions of Theorem 1 (with more general potentials), and we also discuss some related earlier work in the area.

Preparation
In order to make this note as detailed and complete as possible, we shall now review some notions from the theories of obstacle problems and of weighted polynomials. We shall also discuss, in a suitably adapted form, some relevant background from [3].
As general sources, and in particular for detailed proofs of some statements taken for granted below, we refer to the book [17] and the paper [12].
2.1. The obstacle problem. Let F Q be the family of all subharmonic functions f on C which are everywhere ≤ Q and which satisfy f (ζ) ≤ 2 log |ζ| + O(1) as ζ → ∞.
We define a subharmonic functionQ on C by This is the obstacle function corresponding to the obstacle Q; it is well known and easy to check thatQ satisfiesQ ≤ Q andQ(ζ) = 2 log |ζ|+ O(1) as ζ → ∞. Now define the effective potential to be and note that Q eff ≥ 0 on C. By the coincidence set for the obstacle problem we shall mean the compact set It is well known (cf. [12]) thatQ is C 1,1 -smooth on C and harmonic in the complement (S * ) c . Moreover,Q is related to the equilibrium measure σ by is the logarithmic potential of σ. Differentiating in the sense of distributions we have Hence, since S is the support of σ, we have the inclusion In general the difference set S * \ S may be non-empty, consisting then of 'shallow points' in the parlance of [12]. However, our assumption (v) says precisely that there are no shallow points, i.e., that S * = S. Thus condition (v) can be restated as that We will have frequent use for the following simple lemma. A proof is included for completeness.
Here the constant c is any positive number with c < c 0 (cf. (1.9)).
Proof. Fix a boundary point p ∈ ∂S and let N = N p be the unit normal to ∂S at p pointing outwards from S. Let V be a C 2 -smooth (Whitney's) extension of Q eff | S c to a neighbourhood of ∂S. We shall write ∂ N V(p) for the directional derivative in direction N and ∂ T V(p) for the derivative in the (positively oriented) tangential direction to ∂S.
By the C 1,1 -smoothness of Q eff and the fact that Q eff = 0 on S we have For small δ > 0 we hence obtain by Taylor's formula that Finally, by the lower semi-continuity of Q eff and the assumptions (1.1), (2.1) we conclude that Q eff attains a strictly positive minimum over the set {δ(ζ) ≥ δ 0 }. The lemma follows if we denote this minimum value by '2a 0 '.
where q is a holomorphic polynomial of degree at most n − 1.
The following well known lemma is sometimes known as the 'maximum principle of weighted potential theory'. We outline a proof for convenience.
A suitable version of the maximum principle now shows that u ≤Q on C, thus finishing the proof of the lemma.
We now fix, once and for all, an open neighbourhood V of the droplet S, which is small enough so that ∆Q is continuous and strictly positive in a neighbourhood of the closure V. This is possible by assumption (ii).
For a point ζ 0 ∈ V we define the microscopic scale r n = r n (ζ 0 ) as the smallest number r > 0 such that It follows from our assumptions that there are positive numbers k 1 , k 2 such that, for all ζ ∈ V, and all n ≥ n 0 (n 0 large enough) We will require the following lemma, which is of a somewhat similar kind as [3, Lemma 1].

Lemma 2.3.
There are constants s > 0 and C > 0 such that for each weighted polynomial f ∈ W n and each ζ 0 ∈ V such that f (ζ 0 ) 0, we have Proof.
The assumption that f (ζ 0 ) 0 guarantees that g is differentiable at ζ 0 . In the following, we can assume that ζ 0 = 0. Consider the holomorphic polynomial In view of Taylor's formula, we have By (2.3) we obtain that there is a constant s such that Suppose that f = q·e −nQ/2 so that g = |q| 2β e −βnQ . Then, in a neighbourhood of 0, We now introduce the holomorphic functiong := q 2β e −βnH , whose derivative isg Since ∂Q(0) = H ′ (0)/2, we gather from the above that Using a Cauchy estimate, we now deduce that if r is fixed in the interval r n /2 ≤ r ≤ r n , then (writing 'ds' for the arclength measure on the circle |ζ| = r) An integration in r now gives that By this, the lemma is proven.

Random variables.
Let {ζ j } n 1 be a random sample with respect to the Gibbs measure (1.3).
Consider now, for a fixed j with j ∈ {1, . . . , n} the random weighted Lagrange polynomial These weighted polynomials were used in [3] to study the separation of random configurations. We shall here use similar techniques to examine the localization of the gas. Towards this end, let us write X j = X j,n for the random variable The following lemma, whose proof can be found in [3], will play a key rôle in what follows.
Remark. A similar identity was used in the paper [7] to study equidistribution for a class of β-ensembles.

Proof of Theorem 1
We start by fixing a sequence (U n ) ∞ n=1 of bounded open neighbourhoods of the droplet S, which is increasing and exhausts C, viz.
Throughout our argument below we fix an unspecified integer n 0 which can be chosen larger as we go along, and we assume that n ≥ n 0 .
For definiteness, we will fix U n to be the disc of radius log n about the origin, (3.1) U n := D(0; R n ), R n := log n, (n ≥ n 0 ). Now fix j, 1 ≤ j ≤ n, and recall the random variable X j = C |ℓ j | 2β . By Lemma 2.4 we know that We next introduce the two events where λ > 0 is a parameter.
By Chebyshev's inequality and (3.2) we have the basic estimate Passing to complements we conclude that We shall now prove that the probability P β n (A c j ) is 'negligible'.
Lemma 3.1. There are constants h * > 0 and n 0 > 0 such that n ≥ n 0 implies Proof. We will give a proof based on estimates for the partition function which can essentially be found in the union of the papers [12,13]. ( [13] is written in the setting of a real log-gas, but the following argument is virtually the same in the complex case.) We first note, due to the growth assumption (1.1) on Q, that there are numbers h 0 > 0 and n 0 > 0 such that for all ζ, η ∈ C with |η| > R n 0 we have (To see this, use the elementary inequality |ζ − η| 2 ≤ (1 + |ζ| 2 )(1 + |η| 2 ).) Recalling the definition of the partition function Z β n in (1.4), we now write We conclude that there are constants C and h 1 > 0 such that We shall thus be done when we can prove an upper bound of the form Formally, this bound follows from a well known large n expansion of the partition function (see [20]) but we shall here give an elementary proof based on the papers [12,13].
We start with the identity . Now write I := C e −Q and use Jensen's inequality to conclude that the last expression is We must estimate this expression from below. For this, we start by estimating the number where log − x = min{log x, 0}. Now fix a number δ, 0 < δ < 1 so that |t−ζ|<δ log |t − ζ| 2 e −Q(t) dA(t) > −1 for each ζ ∈ C. Then where l δ (ζ) := 2 max{log − |ζ|, log δ}. Now as l δ is bounded and continuous on C we can apply the convergence in (1.10) to obtain for each t ∈ C.
In fact, an examination of the proof of the convergence (1.10) in [12,13] shows that the convergence is uniform convergence in t and in β = β n provided that, say, we have a weak lower bound of the form β · n 2−ε → ∞ for some ε > 0. (More precisely, this follows from the proof of [12, Theorem 2.9]; the crucial point where the condition β ≫ n −2 enters is in the application of [12, Proposition A.1].) Integrating both sides of the limit (3.10) with respect to the measure e −Q(t) dA(t) we infer that the sequence m n is uniformly bounded from below when β ≫ 1/n, which is the case here.
Recalling our standing assumption (vi), we now obtain by (3.8) and (3.9) that a bound of the form (3.7) must hold.
The proof of the lemma is complete.

3).) It follows from Lemma 3.1 and the inequality (3.4) that
It is convenient to briefly discuss the values λ that come into play. We want the term R −βh * n n to be negligible in comparison with the term R 2 n λ −1 , which is the same as requiring that (3.11) log λ ≪ (βh * n + 2) log log n, (n → ∞).
Now recall that we have assumed that βn → ∞ as n → ∞. To be certain that (3.11) is satisfied we shall henceforth only consider λ's that satisfy In the sequel we assume that ζ belongs to a given, small enough open neighbourhood V of S and pick a random sample {ζ j } n 1 . If ζ does not coincide with one of the ζ k 's then by Lemma 2.3 we have the estimate Using the estimate (2.3), we conclude that if B j has occurred then (with a new C) ∇(|ℓ j | 2β ) L ∞ (V) ≤ Ce sβ n 3/2 λ.
Continue to assume that B j has occurred. For any two points ζ, η ∈ S we can pick a smooth curve γ ⊂ V joining them of bounded length where K is some constant depending only on S and V.
We can slightly modify the curve γ so that it avoids the finitely many zeros of ℓ j , at the expense of increasing K by an arbitrarily small amount.
To proceed, we now impose the further restriction on λ that We also fix an arbitrary number α ∈ (0, 1) and choose n 0 large enough that n ≥ n 0 implies For a fixed n ≥ n 0 we now consider the set M n ⊂ S c of points ζ such that Note that if B j has occurred then certainly ζ j M n , for otherwise (3.17) would imply α ≥ |ℓ j (ζ j )| 2β = 1 > α.
Thus with probability at least 1 − ǫ the entire Coulomb gas is actually contained in the neighbourhood M c n of S. To finish the proof we observe that our restrictions on λ are equivalent to that the parameter ǫ (cf. (3.13)) satisfy (3.21) ǫ nR 2 n e −na 0 and ǫ nR 2 n e −βh * n/2 . For such ǫ we now write The inequality (3.20) is then equivalent to that which transforms to Thus if we define so the Coulomb gas is with probability at least 1 − ǫ contained in the δ nneighbourhood of S. In symbols, we have shown that P β n ({D n ≥ δ n }) ≤ ǫ. If we write ǫ = e −t and µ n = log ν n + log(C/α) + sβ this becomes Now notice that log ν n ≍ log log n as n → ∞, so µ n ≍ log log n + β. We have shown (3.22) under the hypothesis (3.21). Since we have assumed that β · n → ∞ as n → ∞, (3.21) surely holds if t ≤ aβn for a small enough a > 0.
The proof of Theorem 1 is complete. QED

Concluding remarks
In this section, shall gradually generalize Theorem 1, by allowing for more and more general potentials. After that, we will comment on related results and say something about future prospects.

Bulk singularities.
It is useful to allow for potentials giving rise to points ζ in the bulk Int S at which ∆Q(ζ) = 0. Such points have been termed bulk singularities, cf. [5] and the references there.
(Example: The droplet of the Mittag-Leffler potential Q = |ζ| 2λ is the closed disc about the origin of radius λ −1/2λ . If λ > 1 then clearly the origin is a bulk singularity.) A careful inspection of our previous arguments shows that the crucial property we need is that the number c 0 = min ∂S {∆Q} be strictly positive, i.e., it suffices instead of assumption (ii) to assume that (ii') Q is strictly subharmonic in a neighbourhood of the boundary ∂S.
In order to generalize our main result, we would foremost like to generalize Lemma 2.3 to a setting with bulk singularities.
To be precise, we want to prove that if V is a small enough open neighbourhood of S then for each weighted polynomial f = q · e −nQ/2 ∈ W n and each point ζ 0 ∈ V at which f (ζ 0 ) 0, we have the estimate Here s is a suitable constant and r n (ζ 0 ) is the microscopic scale defined in (2.2). In order to obtain (4.1) near a bulk singularity, it turns out that we require a higher regularity of the potential, i.e., C 2 -smoothness will not quite suffice near such a point.
For this and other reasons it will be convenient in the following to simply assume that: (vii) Q is real-analytic in Int{Q < +∞}.
Remark. Referring to our standing assumptions, the condition (iii) is automatically implied by (vii) and our other assumptions, by virtue of Sakai's regularity theorem (cf. [14] and references). Sakai's theorem moreover gives that (for potentials real-analytic near the boundary of S) ∂S is the union of finitely many real-analytic arcs and possibly finitely many singular points. Assumption (iv) precludes the appearance of any singular points.
In the proof of Lemma 2.3 we replace the second-degree polynomial H by It is then easy to check that we still have nβ|Q(ζ) − Re H(ζ)| ≤ sβ when |ζ| ≤ r n , where s is a suitable constant, i.e., we have the property (2.5).
The rest of our proof of Lemma 2.3 now works virtually the same, proving (4.1).
We next observe that if the neighbourhood V of S is sufficiently small then r n (ζ) 1/ √ n with a constant that is uniform for all ζ ∈ V. (See (2.2).) Combining this with (4.1) we have shown that (4.2) |∇g(ζ 0 )| e sβ n 3/2 C g dA.
Using the estimate (4.2) instead of Lemma 2.3, the rest of our proof of Theorem 1 works virtually the same.
We summarize our findings in the form of a theorem.

4.2.
Perturbations of real-analytic potentials. Let us fix a potential Q obeying the various conditions in the preceding subsection; in particular Q is assumed to be real-analytic in a neighbourhood of S. It is well known that, for C ∞ -smooth potentials, the droplet may be highly irregular, cf. [18]. Fortunately, the class of n-dependent small perturbations Q + u/n of real-analytic potentials Q is adequate to deal with, basically, all cases of interest. On the other hand, it is of interest to allow the perturbation u/n to be as general as possible. To keep it simple, we shall here consider smooth perturbations only.
With this in mind, we now fix a bounded, C 2 -smooth function u and and consider the n-dependent potential In a 'classical' meaning, the potentials Q and V n are indistinguishable; their droplets and equilibrium measures are the same. The difference between them appears on the statistical level, when we introduce the Gibbs measure corresponding to V n , dP β n ∝ e −βH n dA n , H n := n j k More precisely, the weakly n-dependent term (u/n) affects the distribution of particles near the boundary.
The preseant more general situation can be treated similarly as before, by redefining the class W n of weighted polynomials to consist of elements of the form where q is a holomorphic polynomial of degree at most n − 1. Again we consider this as a subspace of L 2 .
In the present case Lemma 2.2 must be modified to where C is a constant depending on the perturbation u. The rest of the proof works virtually the same as in the earlier cases.
We hence obtain without difficulty the following generalization of Theorem 2.

Theorem 3.
Under the above assumptions, the result of Theorem 1 remains true when {ζ i } n 1 is picked randomly with respect to the potential V n . 4.3. Comparison with earlier results. Suppose that β = 1, Q is radially symmetric, and S is a disc of radius R. In this case we have recognized R+D n as, essentially, the spectral radius of a matrix picked randomly from a certain normal matrix ensemble. The distribution of this spectral radius was worked out by Rider [16] for the Ginibre ensemble (the potential Q = |ζ| 2 ) and Chafaï and Péché for more general radially symmetric potentials [8].
The results in [8,16] imply that ω n converges in distribution to the standard Gumbel distribution as n → ∞, where γ n = log(n/2π) − log log n + log(R 2 c 0 ).
(We may recall here that a random variable X is said to have a standard Gumbel distribution if its distribution function is P(X ≤ t) = exp(− exp(−t)).) The theorems of Rider and Chafaï-Péché can be said give a kind of twodimensional analogue to the well known convergence to the Tracy-Widom distribution for the top eigenvalue in Hermitian random matrix theory [1,19]. In this connection, it is interesting to recall that a very precise asymptotic for the Tracy-Widom β-distribution for certain one-dimensional ensembles was worked out by Dumaz and Virág in the paper [10].
To further compare with our Theorem 1 we observe that as n → ∞ This is of the same order of magnitude ( log n n ) as our present bound in Theorem 1. Incidentally, we see that our value for the constant A of proportionality in (1.11) can be improved by a factor 1/ √ 5 in this case. Note that even in the case when β = 1 and Q radially symmetric, it can happen that the droplet has several boundary components. For example, if Q(ζ) = |ζ| 2 + log(1/|ζ| 2 ) then the droplet S is the annulus {1 ≤ |ζ| ≤ √ 2}. The random variable D n of course measures the largest distance from a ζ j to the droplet, irrespective of which boundary component happens to be closer. Thus, of course, D n can not be regarded as a spectral radius in this case.
We next recall a few facts concerning the case of temperature zero (or inverse temperature β = ∞). In this case, it is customary to interpret {ζ j } n 1 to be a weighted Fekete configuration, i.e., it is defined to be a minimizer of the weighted energy in (1.2). Fekete-configurations are wholly contained in the droplet, viz., we have exactly D n = 0 at β = ∞ for each finite n. (See [17] for a proof of this.) Fekete points are equidistributed in the droplet with respect to the equilibrium measure σ, see e.g. the references in [3] as well as [15].
The low temperature regime when β ≍ log n was studied in [3]. In such a setting, our present results show that the gas is effectively localized to a microscopic neighbourhood of S, i.e., to a neighbourhood of the form {ζ; δ(ζ) n −1/2 }.
Due to limitations of our methods, we do not seem to quite reach up to high temperatures of the magnitude β ≍ 1/n here. This kind of regime is however studied, in a suitably adapted setting, in the recent paper [2].
To end this note, I want to briefly point to the self-improving method from the paper [4]. This method was developed with a partial intention to eventually obtain a rigorous proof of full plane Gaussian field convergence of linear statistics of a Coulomb gas, but due to some technical challenges it was only applied when β = 1. (The influential paper [13] provides a somewhat analogous construction on R, which was also applied to βensembles.) One of the technical obstacles for extending the proof to cover β-ensembles involved having a good enough decay of the 1-point function in the exterior of the droplet, which is essentially what we have done here for all β > 0. This is not the right place to elaborate on the problem of proving suitable interior estimates in our setting. Nevertheless, I want to point out that the papers [6,15] provide judicious analyses in this direction.