Repulsion in low temperature $\beta$-ensembles

TL;DR

This paper proves particle separation in a 2D Coulomb plasma at inverse temperatures greater than 1, showing that for large $eta$, particles are separated at the scale of the conjectured Abrikosov lattice.

Contribution

It establishes a new result on particle separation in Coulomb plasmas for $eta>1$, connecting to the conjectural Abrikosov lattice structure.

Findings

Particle separation occurs for $eta>1$

Separation scale matches Abrikosov lattice conjecture at large $eta$

Provides mathematical proof for particle distribution in Coulomb systems

Abstract

We prove a result on separation of particles in a two-dimensional Coulomb plasma, which holds provided that the inverse temperature $\beta$ satisfies $\beta>1$. For large $\beta$, separation is obtained at the same scale as the conjectural Abrikosov lattice optimal separation.

Equations

$$d{\mathbf{P}}_n^{(\beta )}(\zeta )=\frac{1}{Z_n^{(\beta )}}{e}^{-\beta H_n(\zeta )}d{A}^{\otimes n}(\zeta ),\zeta =({\zeta }_1,\dots ,{\zeta }_n)\in {\mathbb{C}}^n.$$

$$H_n\left({\zeta }_1,\dots ,{\zeta }_n\right)=\sum _{j\ne k}\log \frac{1}{|{\zeta }_j-{\zeta }_k|}+n\sum _{j=1}^{n}Q({\zeta }_j),$$

$$I_Q[\mu ]={\iint }_{{\mathbb{C}}^2}\log \frac{1}{|\zeta -\eta |}d\mu (\zeta )d\mu (\eta )+{\int }_{\mathbb{C}}Qd\mu .$$

$$d\sigma (\zeta )={\chi }_S(\zeta )\Delta Q(\zeta )dA(\zeta ),$$

$$n{\int }_{{\mathbb{D}}_{r_n}}\Delta QdA=1.$$

$$z_j=r_n^{-1}{\zeta }_j.$$

$$s_0=\underset{z_j\in \mathbb{D}}{\min }\underset{k\ne j}{\min }|z_j-z_k|,(\{z_j{\}}_1^n\in {\mathcal{F}}_n).$$

$${\mathbb{P}}_n^{(\beta )}(\{s_0\ge c\cdot n^{-\frac{1}{\beta -1}}\cdot (ϵ\eta {)}^{\frac{1}{2(\beta -1)}}\}|{\mathcal{F}}_n)\ge 1-m_0ϵ,$$

$${\eta }_n\ge const.n^{-2\vartheta },(const.>0).$$

$$\Delta Q(0)\ge const.>0.$$

$$\underset{\beta \to \infty }{\lim }{\mathbb{P}}_n^{(\beta )}\left(\left\{s_0>c\right\}|{\mathcal{F}}_n\right)=1.$$

$$\underset{n\to \infty }{\lim }\underset{\beta \ge \mu \log n}{\inf }{\mathbb{P}}_n^{(\beta )}\left(\left\{s_0>c{e}^{-(1+\vartheta )/\mu }\right\}|{\mathcal{F}}_n\right)=1.$$

$$\Delta Q=P+\text{"higherorder terms”}$$

$${\tau }_0^{-2k}=\frac{1}{2\pi k}{\int }_0^{2\pi }P(e^{i\theta })d\theta .$$

$${\tau }_0^{-2k}=\frac{{\Delta }^{k}Q(0)}{k[(k-1)!{]}^2}.$$

$$r_n={\tau }_0{n}^{-1/2k}(1+O(n^{-1/2k})),(n\to \infty ).$$

$$H(\zeta ):=Q(0)+2\partial Q(0)\zeta +\cdots +\frac{2}{(2k)!}{\partial }^{2k}Q(0){\zeta }^{2k}.$$

$$Q_0(\zeta )=\sum _{i+j=2k,i,j\ge 1}\frac{{\partial }^i{\bar{\partial }}^{j}Q(0)}{i!j!}{\zeta }^i{\bar{\zeta }}^j.$$

$$Q(\zeta )=\mathrm{Re}H(\zeta )+Q_0(\zeta )+O(|\zeta {|}^{2k+1}),(\zeta \to 0).$$

$$|\nabla |f|(0)|\le K{r}_n^{-1}{\text{fint}}_{{\mathbb{D}}_{r_n}}|f|.$$

$$q_0=\sum _{i+j=2k,i,j\ge 1}\frac{|{\partial }^i{\bar{\partial }}^{j}Q(0)|}{i!j!}.$$

$$|\zeta |\le r_n\Rightarrow n|Q(\zeta )-h(\zeta )|\le q_{0}n|\zeta {|}^{2k}+O(n^{-1/2k})\le C_n,$$

$$|\nabla |f|(\zeta )|=|p^{\prime }(\zeta )-n\partial Q(\zeta )p(\zeta )|e^{-nQ(\zeta )/2}$$

$$\left|\nabla \left(|p|e^{-nh/2}\right)(\zeta )\right|=\left|\frac{d}{d\zeta }\left(p{e}^{-nH/2}\right)(\zeta )\right|.$$

$$|\nabla |f|(0)|=\left|\frac{d}{d\zeta }\left(p{e}^{-nH/2}\right)(0)\right|.$$

$$\left|\frac{d}{d\zeta }\left(p{e}^{-H/2}\right)(0)\right|=\frac{1}{2\pi }\left|{\int }_{|\zeta |=r}\frac{p(\zeta )e^{-nH(\zeta )/2}}{{\zeta }^2}d\zeta \right|\le \frac{2}{\pi r_n^2}{\int }_{|\zeta |=r}|p|e^{-nh/2}|d\zeta |.$$

$$e^{C_n/2}{\int }_{|\zeta |=r}|p|e^{-nQ/2}|d\zeta |=e^{C_n/2}{\int }_{|\zeta |=r}|f||d\zeta |.$$

$$|\nabla |f|(0)|$$

$${\ell }_j(\zeta )=\left(\prod _{i\ne j}(\zeta -{\zeta }_i)/\prod _{i\ne j}({\zeta }_j-{\zeta }_i)\right)\cdot e^{-n(Q(\zeta )-Q({\zeta }_j))/2},(j=1,\dots ,n).$$

$${\mathbf{E}}_n^{(\beta )}\left[{\chi }_U({\zeta }_j)\cdot {\int }_{\mathbb{C}}|{\ell }_j(\zeta ){|}^{2\beta }dA(\zeta )\right]=|U|.$$

Full text

Consider a large but finite system of identical point-charges $\{\zeta_{i}\}_{1}^{n}$ in the plane ${\mathbb{C}}$, in the presence of an external field $nQ$, such that $Q(\zeta)$ is "large” near $\zeta=\infty$. The system is picked randomly from the Boltzmann-Gibbs distribution at inverse temperature $\beta>1$,

[TABLE]

Here $H_{n}$ is the total energy

[TABLE]

$dA=dxdy/\pi$ is Lebesgue measure on ${\mathbb{C}}$ divided by $\pi$. The constant $Z_{n}^{(\beta)}=\int e^{-\beta H_{n}}\,dA^{\otimes n}$ is the so-called partition function of the ensemble.

A random sample $\{\zeta_{j}\}_{1}^{n}$ might be termed "Coulomb gas”, "one-component plasma”, or "$\beta$-ensemble”. For brevity, we use "system” as a synonym.

It is well-known that the system tends, on average, to follow Frostman’s equilibrium measure in external potential $Q$. The support of the equilibrium measure is a compact set which we call the droplet.

The rough approximation afforded by the equilibrium measure is too crude to reveal details on a microscopic scale. However, it is believed on physical grounds that the particles should be evenly spread out in the interior of the droplet, with a non-trivial behaviour near the boundary - the Hall effect. Everything of importance goes on in the vicinity of the droplet.

In this note, we prove that the distance between neighbouring particles at a given location in the plane is large with high probability. Further, the distance tends to increase with $\beta$, and as $\beta\to\infty$, we recover formally the separation theorem for Fekete sets from the papers [1, 5].

Formulation of results

Let $Q:{\mathbb{C}}\to{\mathbb{R}}\cup\{+\infty\}$ be a suitable function of sufficient increase near $\infty$; precise conditions are given below. We call $Q$ the external potential.

Let $\mu$ be a compactly supported Borel probability measure on ${\mathbb{C}}$. The weighted logarithmic energy of $\mu$ is defined by

[TABLE]

Assuming that $Q$ obeys some natural conditions recalled below there is a unique compactly supported probability measure $\sigma$ which minimizes $I_{Q}$. This is Frostman’s equilibrium measure in external potential $Q$. The support $S=\operatorname{supp}\sigma$ is known as the droplet, and the equilibrium measure takes the form (see [26])

[TABLE]

where we write $\Delta={\partial}\bar{\partial}$ for $1/4$ times the standard Laplacian; $\chi_{S}$ is the characteristic function of the set $S$.

Remark*.*

Let $\{\zeta_{j}\}_{1}^{n}$ be a random sample with respect to $\mathbf{P}_{n}^{(\beta)}$. Write ${\mathbf{E}}_{n}^{(\beta)}$ for the expectation with respect to $\mathbf{P}_{n}^{(\beta)}$. It is well-known that ${\mathbf{E}}_{n}^{(\beta)}[\frac{1}{n}\sum_{j=1}^{n}f(\zeta_{j})]\to\sigma(f)$ as $n\to\infty$ for each continuous bounded function $f$ on ${\mathbb{C}}$. See [18, 21].

The preceding remark shows that, in a sense, the equilibrium measure gives a first approximation to the macroscopic behaviour of the system. We here want to study microscopic properties. For this, we could fix a point $p\in{\mathbb{C}}$, which might depend on $n$, and zoom on it at an appropriate rate. However, for technical reasons it is easier to choose the coordinate system so that $p=0$. In other words, [math] will in the following denote the origin of an $n$-dependent coordinate system which can be obtained from some static reference system by rigid motion.

Let ${\mathbb{D}}_{r}={\mathbb{D}}_{r}(0)$ denote the disk center [math] radius $r$. By the microscopic scale $r_{n}$ at [math] we mean the radius such that

[TABLE]

We allow for any situation such that $r_{n}$ is well-defined. This is a mild restriction. Indeed, we always have $\Delta Q\geq 0$ on $S$, since $\sigma$ is a probability measure. By our assumptions below, this implies that $r_{n}$ is always well-defined if [math] is in the interior of $S$. Also, if we have $\Delta Q>0$ on some portion of ${\partial}S$, then $r_{n}$ is well-defined when [math] is in some neighbourhood of that portion. Since the behaviour of the gas is of interest only in a neighbourhood of the droplet, we can thus essentially treat all cases of interest.

Given a sample $\{\zeta_{j}\}_{1}^{n}$, we rescale about [math] and consider the process $\{z_{j}\}_{1}^{n}$ where

[TABLE]

We denote by ${\mathbb{P}}_{n}^{(\beta)}$ the image of $\mathbf{P}_{n}^{(\beta)}$ under the map (1) and write ${\mathbb{E}}_{n}^{(\beta)}$ for the corresponding expectation. Also let ${\mathbb{D}}={\mathbb{D}}_{1}$ be the unit disk.

Fix a large $n$ and let ${\mathscr{F}}_{n}$ be the event that at least one of the $z_{j}$ falls in ${\mathbb{D}}$. Denote $\eta=\eta_{n}={\mathbb{P}}_{n}^{(\beta)}({\mathscr{F}}_{n})$.

Given a random sample $\{z_{j}\}_{1}^{n}\in{\mathscr{F}}_{n}$ we define a number $s_{0}$ by

[TABLE]

Thus $s_{0}$ is the largest rescaled distance from a particle in ${\mathbb{D}}$ to its nearest neighbour. We refer to $s_{0}$ as the spacing of the sample, in the vicinity of the point [math].

We are now prepared to formulate our main results. The following result shows that the strength of repulsion tends to increase with $\beta$.

Theorem**.**

Suppose that $\beta>1$ and fix $n_{0}\geq 1$. Then there is a constant $c=c(n_{0},\beta)>0$ such that if $n\geq n_{0}$ and $0<\epsilon<1$, then

[TABLE]

*where $m_{0}=16n^{\frac{2}{\beta-1}}c^{-2}(\epsilon\eta)^{-\frac{1}{\beta-1}}$. Moreover, given any $\beta_{0}>1$, $c$ can be chosen independent of $\beta$ when $\beta\geq\beta_{0}$. *

The left hand side in (2) should be understood as a conditional probability given that ${\mathscr{F}}_{n}$ has occurred.

The next result gives a kind of separation which holds for large $\beta$. To this end, it is natural to assume some kind of lower bound on the probability $\eta_{n}$. One possibility is to assume that that $\inf\eta_{n}>0$, which is certainly a reasonable assumption in many cases. However, it will suffice to assume existence of some number $\vartheta\geq 0$ such that

[TABLE]

For simplicity, we will also assume that we are zooming on a regular point,

[TABLE]

Corollary**.**

Below fix a positive number $c$ with $c<1/(8\sqrt{e})$.

1. (i)

Suppose that (4) holds and let $n$ be a given large integer. Then

[TABLE] 2. (ii)

Suppose that (3) and (4) hold. Also fix a parameter $\mu>0$. Then

[TABLE]

Condition (3) is reasonable when the droplet is "sufficiently present" at [math], see concluding remarks.

Case (i) of the corollary comes close to an unpublished result due to Lieb in the zero temperature case, see [24, Theorem 4] as well as [25]; cf. [5] for an independent proof. Our estimate for the constant $c$ should be compared with the asymptotic lower bound $1/\sqrt{e}$ for the distance between Fekete points obtained in [1, Theorem 1]. In fact, our method of proof is somewhat related to the approach in [1, 5], see concluding remarks below.

In the present context, Abrikosov’s conjecture states that under the conditions in Corollary, the system $\{z_{j}\}_{1}^{n}$ should more and more resemble a honeycomb lattice as $\beta\to\infty$. The distance between neighbouring particles in this lattice can be computed, leading naturally to the conjecture that the "right” bound for $c$ in Corollary should be $c<2^{1/2}3^{-1/4}$. Cf. [1].

Here are precise assumptions to be used in the proofs below: (i) $Q:{\mathbb{C}}\to{\mathbb{R}}\cup\{+\infty\}$ is l.s.c.; (ii) the interior of the set $\Sigma:=\{Q<\infty\}$ is non-empty; (iii) $Q$ is real-analytic on $\operatorname{Int}\Sigma$; (iv) $\liminf_{\zeta\to\infty}Q(\zeta)/\log|\zeta|^{2}>1$; (v) $S\subset\operatorname{Int}\Sigma$.

In addition, we freely use the following notation: The $dA$-measure of a subset $\omega\subset{\mathbb{C}}$ is denoted $|\omega|$. By ${\mathscr{W}}_{n}$ we mean the set of weighted polynomials $f=pe^{-nQ/2}$ where $p$ is a holomorphic polynomial of degree at most $n-1$. We denote averages by $\fint_{\omega}g=\frac{1}{|\omega|}\int_{\omega}g\,dA$. ${\mathbb{D}}_{r}(\zeta)$ denotes the disc center $\zeta$ radius $r$ and we write ${\mathbb{D}}_{r}={\mathbb{D}}_{r}(0)$.

Proofs of the main results

Suppose that the Taylor expansion of $\Delta Q$ about [math] takes the form

[TABLE]

where $P\geq 0$, $P\not\equiv 0$, and $P$ is homogeneous of some degree $2k-2$. The existence of such a $P$ is of course a consequence of the real-analyticity of $Q$.

Following [7] we write $\tau_{0}$ for the positive constant such that

[TABLE]

Note that $\tau_{0}$ can be cast in the form

[TABLE]

This follows easily by expressing $P$ as a polynomial in $\zeta$ and $\bar{\zeta}$.

Using $\tau_{0}$, we conveniently express the microscopic scale to a negligible error, as follows

[TABLE]

Note that if $k=1$ then $\tau_{0}=1/\sqrt{\Delta Q(0)}$.

As in [7] we define a holomorphic polynomial $H$ by

[TABLE]

We will also use the dominant homogeneous part of $Q$ at [math], i.e., the function

[TABLE]

The point is that we have the canonical decomposition (cf. [7])

[TABLE]

Below we fix a large integer $n_{0}$. The following Bernstein-type lemma is an elaboration of [1, Lemma 2.1].

Lemma 1**.**

Suppose that $n\geq n_{0}$. If $f\in{\mathscr{W}}_{n}$ and $f(0)\neq 0$ then there is a constant $K=K(n_{0})$ such that

[TABLE]

If $\Delta Q(0)\geq\mathrm{const.}>0$ then we can take $K(n_{0})=4\sqrt{e}(1+o(1))$, $(n_{0}\to\infty)$.

Proof.

Denote $h=\operatorname{Re}H$ where $H$ is the polynomial in (5). Also write

[TABLE]

Since $r_{n}=\tau_{0}n^{-1/2k}(1+O(n^{-1/2k}))$ we have

[TABLE]

where $C_{n}=\tau_{0}^{2k}q_{0}+Cn^{-1/2k}.$

Now note that

[TABLE]

and

[TABLE]

Inserting $\zeta=0$ it is now seen that

[TABLE]

We now apply a Cauchy estimate to deduce that if $r_{n}/2\leq r\leq r_{n}$ then

[TABLE]

By (6) the last integral is dominated by

[TABLE]

It follows that

[TABLE]

where $K=4\sup_{n\geq n_{0}}\{e^{C_{n}/2}\}$. If $\Delta Q(0)\geq\mathrm{const.}>0$ then $k=1$ and $\tau_{0}^{2}q_{0}=1$, which gives $K\leq 4e^{1/2+C/\sqrt{n_{0}}}$. ∎

The weighted Lagrange interpolation polynomials associated with a configuration $\{\zeta_{j}\}_{1}^{n}$ of distinct points are defined by

[TABLE]

Note that $\ell_{j}\in{\mathscr{W}}_{n}$ and $\ell_{j}(\zeta_{k})=\delta_{jk}$.

Now let $\{\zeta_{j}\}_{1}^{n}$ be a random sample from $\mathbf{P}_{n}^{(\beta)}$. Then $\ell_{j}(\zeta)$ is a random variable which depends on the sample and on $\zeta$. In the next few lemmas, we fix an index $j$, $1\leq j\leq n$.

Lemma 2**.**

Suppose that $U$ is a measurable subset of ${\mathbb{C}}$ of finite measure $|U|$. Then

[TABLE]

Proof.

We shall use the following identity, whose verification is left to the reader

[TABLE]

By this and Fubini’s theorem, integrating first in $\zeta_{j}$, we get

[TABLE]

proving (7). ∎

In the sequel, we assume that $n\geq n_{0}$ and recall the constant $K=K(n_{0})$ provided by Lemma 1. We will write $K(n_{0},\zeta)$ the same constant with [math] replaced by $\zeta$ and let $r_{n}(\zeta)$ be the microscopic scale at $\zeta$. Finally, we fix a suitable, large enough, constant $M$; we may take $M=3$ for example.

It is easy to see that there is a constant $T=T(M,n_{0})\geq 1$ such that if $\zeta\in{\mathbb{D}}_{Mr_{n}}$ and $n\geq n_{0}$ then $T^{-1}r_{n}(\zeta)\leq r_{n}(0)\leq Tr_{n}(\zeta)$. If $\Delta Q(0)\geq\mathrm{const.}>0$ we might take $T=1+o(1)$ as $n_{0}\to\infty$.

Lemma 3**.**

We have that

[TABLE]

Now suppose that $\beta\geq 1/2$, $n\geq n_{0}$, $K=\sup_{\zeta\in{\mathbb{D}}_{Mr_{n}}}K(\zeta)$, and $r_{n}=r_{n}(0)$. Then

[TABLE]

Proof.

The identity (8) follows from Lemma 2 with $U={\mathbb{D}}_{r_{n}}$.

To prove (9) we fix a non-zero $f\in{\mathscr{W}}_{n}$ and assume that $f(\zeta)\neq 0$ where $\zeta\in{\mathbb{D}}_{Mr_{n}}$. By Lemma 1 and Jensen’s inequality, we have for all $\beta\geq 1/2$ that

[TABLE]

Applying this with $f=\ell_{j}$ and taking expectations, we get

[TABLE]

where we used (8) in the last step. ∎

In the following, we let $z$ and $\zeta$ denote complex variables related via

[TABLE]

We shall use the random functions ${\varrho}_{j}$ defined by

[TABLE]

Thus ${\varrho}_{j}(z_{k})=\delta_{jk}$ where $\{z_{k}\}_{1}^{n}$ is the rescaled process.

Lemma 4**.**

Let $\beta\geq 1/2$. Then with notation as above

[TABLE]

Proof.

The inequality (9) says that

[TABLE]

Rescaling we immediately obtain (10). ∎

Suppose that $\beta>1$. We will use Morrey’s inequality, which asserts that for all real-valued $f$ in the Sobolev space $W^{1,2\beta}({\mathbb{D}}_{M})$, all $z,w\in{\mathbb{D}}_{M/\sqrt{2}}$, we have

[TABLE]

See [10, Corollary 9.12] and its proof. In fact, the proof in [10, p. 283] shows that (11) holds with $C=C_{0}(1-1/\beta)^{-1}$ where $C_{0}\leq 2\pi^{1/2\beta}$.

We are now ready to finish the proof of Theorem.

Recall that ${\mathscr{F}}_{n}$ denotes the event that at least one particle hits ${\mathbb{D}}$ and fix an arbitrary $j$, $1\leq j\leq n$. Assuming that ${\mathbb{P}}_{n}^{(\beta)}({\mathscr{F}}_{n})\geq\eta>0$, we deduce from Lemma 4 the following inequality for the conditional expectation

[TABLE]

Fix $\epsilon>0$ and recall that $s_{0}$ denotes the distance from a point in $\{z_{j}\}_{1}^{n}\cap{\mathbb{D}}$ to its closest neighbour, where we assume that $\{z_{j}\}_{1}^{n}\in{\mathscr{F}}_{n}$. We must prove that $s_{0}\geq c(\epsilon\eta)^{1/2(\beta-1)}$ with (conditional) probability at least $1-\epsilon$.

For each $\lambda>0$ we have by Chebyshev’s inequality and (12)

[TABLE]

which implies

[TABLE]

Given a random sample $\{z_{j}\}_{1}^{n}\in{\mathscr{F}}_{n}$ we let $I_{n}=I_{n}(\{z_{j}\}_{1}^{n})$ be the random, nonempty set of indices $j$ for which $z_{j}\in{\mathbb{D}}$; we then have

[TABLE]

We now set

[TABLE]

Consider the event $A_{n}$ consisting of all samples $\{z_{j}\}_{1}^{n}\in{\mathscr{F}}_{n}$ such that there is a $j\in I_{n}(\{z_{j}\}_{1}^{n})$ for which $\|\nabla{\varrho}_{j}\|_{L^{2\beta}({\mathbb{D}}_{M})}^{\,2\beta}>n\lambda$. By (13) and our choice of $\lambda$ we have ${\mathbb{P}}_{n}^{(\beta)}(A_{n}|{\mathscr{F}}_{n})\leq\epsilon$. Hence, with conditional probability at least $1-\epsilon$,

[TABLE]

Now fix a sample $\{z_{j}\}_{1}^{n}\in{\mathscr{F}}_{n}$ and an index $j\in I_{n}(\{z_{j}\}_{1}^{n})$. Let $z_{k}$ be a closest neighbour to $z_{j}$. By Morrey’s inequality (11) and (14) there is another constant $C=C_{0}(1-1/\beta)^{-1}$ such that, with conditional probability at least $1-\epsilon$, we have either $|z_{j}-z_{k}|\geq M/\sqrt{2}$ or

[TABLE]

i.e., $|z_{j}-z_{k}|\geq cn^{-\frac{1}{\beta-1}}(\epsilon\eta)^{\frac{1}{2(\beta-1)}}$ where we may chose

[TABLE]

We have shown that

[TABLE]

with probability at least $1-\epsilon$. The formula (16) shows that $c$ can be chosen independent of $\beta$ when $\beta\geq\beta_{0}>1$.

Lemma 5**.**

*Let $N_{\mathbb{D}}$ be the number of particles which fall in ${\mathbb{D}}$. Also define $r_{0}=cn^{-\frac{1}{\beta-1}}(\epsilon\eta)^{\frac{1}{2(\beta-1)}}/2$ and $m_{0}=4/r_{0}^{2}$. Then $N_{\mathbb{D}}\leq m_{0}$ and $s_{0}\geq cn^{-\frac{1}{\beta-1}}(\epsilon\eta)^{\frac{1}{2(\beta-1)}}$ with conditional probability at least $1-m_{0}\epsilon$. *

Proof.

Suppose that at least $m$ particles, denoted $z_{1},\ldots,z_{m}$, fall in ${\mathbb{D}}$. For $1\leq j\leq m$ let $E_{j}$ be the event that the disk ${\mathbb{D}}_{r_{0}}(z_{j})$ contains no point $z_{k}$ with $1\leq k\leq n$, $k\neq j$. Then ${\mathbb{P}}_{n}^{(\beta)}(E_{j}^{c}|{\mathscr{F}}_{n})\leq\epsilon$, where $E_{j}^{c}$ is the complementary event, so

[TABLE]

It follows that if at least $m$ particles fall in ${\mathbb{D}}$ then with probability at least $1-m\epsilon$ there are $m$ disjoint disks of radius $r_{0}$ inside ${\mathbb{D}}_{2}$. Comparing areas we see that $mr_{0}^{2}\leq 4$, i.e., $m\leq m_{0}$. ∎

The lemma says that if $m_{0}=m_{0}(\beta,\epsilon,\eta,n)=16n^{\frac{2}{\beta-1}}c^{-2}(\epsilon\eta)^{-\frac{1}{\beta-1}}$ then

[TABLE]

This proves Theorem.

Now assume that $\Delta Q(0)\geq\mathrm{const.}>0$. Then by Lemma 1, the constant $K$ there might be taken as $K=4\sqrt{e}(1+o(1))$ while we may take $T=1+o(1)$ as $n\to\infty$. Hence $C_{0}T^{1+2/\beta}K\leq 8\sqrt{e}(\pi T^{4})^{1/2\beta}(1+o(1))$, and so, if $c(\beta)$ denotes the largest constant such that the estimate (17) holds asymptotically, as $n_{0}\to\infty$, then

[TABLE]

Applying the assumptions that $\eta\geq\mathrm{const.}n^{-2\vartheta}$ and $\beta\geq\mu\log n$ we deduce that (as $n\to\infty$)

[TABLE]

and

[TABLE]

It is now clear from (18) that Corollary is a consequence of Theorem.

q.e.d.

Concluding remarks

It is natural to ask for conditions implying that the probabilities $\eta_{n}={\mathbb{P}}_{n}^{(\beta)}({\mathscr{F}}_{n})$ satisfy something like

[TABLE]

In the case $\beta=1$, the $\eta_{n}$ are bounded below if $\liminf_{n\to\infty}r_{n}^{2}|{\mathbb{D}}_{r_{n}}\cap S|>0$. ("The proportion of the area of ${\mathbb{D}}_{r_{n}}$ which falls inside the droplet is bounded below.") Proofs depending on estimates for the Bergman function can be found in [3, 4, 6]. On the other hand, if [math] is well in the exterior, or if [math] is a singular boundary point, $\eta_{n}$ drops off to zero quickly as $n\to\infty$. It is natural to expect a similar behaviour for any given $\beta$.

The analysis of Fekete configurations in [1, 5] depends on the inequality $|\ell_{j}|\leq 1$ for the associated weighted Lagrange polynomials $\ell_{j}$. This bound plays a similar role when $\beta=\infty$ as the $L^{2\beta}$-estimate in Lemma 2 does in the present case. The idea of using an $L^{2\beta}$-bound on Lagrange sections occurs in [14]. The context there is different, but in a way, we have elaborated on this idea here.

When $\beta=1$, limiting point fields $\{z_{j}\}_{1}^{\infty}$ have been identified in many cases, [3]. When $\beta>1$, the determinantal structure is lost, and the problem of calculating limiting point fields remains a challenge. The question is perhaps especially intriguing when we rescale about a regular boundary point of $S$, or about some other kind of special point, cf. [4, 6, 23].

At a regular boundary point, it seems plausible that the distribution should be translation invariant in the direction tangent to the boundary, i.e., in a suitable coordinate system, the distribution depends only on $x=\operatorname{Re}z$. The Hall effect is believed to give rise to certain irregularities in the distribution, which are to be located slightly to the inside of the boundary, see [12]. While our results provide more and more information when $\beta$ gets very large, the results in [12], by contrast, seem to be more accurate when $\beta$ is close to $1$. A corresponding analysis was performed earlier in the bulk in [19]; see [13, 15, 20] for more recent developments.

In the case of "moderately sized” $\beta$, $1\ll\beta\ll\infty$, neither of the methods seem to give very clear pictures of the situation. However, the recent paper [16] gives some results for the case $\beta=2$. Moreover, the paper [11] suggests that a phase-transition ("freezing”) should take place after a certain finite value $\beta=\beta_{0}$. The study of existence and possible size of melting temperature $1/\beta_{0}$ is currently an active area of research.

By the "hard edge $\beta$-ensemble” in external potential $Q$, we mean the ensemble obtained by redefining $Q$ to be $+\infty$ outside of the droplet. Cf. [3, 4, 27] for the case $\beta=1$. The question of spacings in this setting will be taken up elsewhere.

Ward’s identity (or "loop equation", "fundamental relation") is a relation connecting the one- and two-point functions of a $\beta$-ensemble. In the present context, it was used systematically by Wiegmann and Zabrodin and their school, and it is an important tool in conformal field theory (CFT). In fact a whole family of Ward identities is known, see [22].

In the paper [2], Ward’s identity was used to give a relatively simple proof of Gaussian field convergence of linear statistics of a $\beta=1$ ensemble. A similar statement is believed to hold for general $\beta$-ensembles. There has been progress on $\beta$-ensembles recently: the paper [8] seems to prove Gaussian field convergence in the bulk of the droplet. To the best of our knowledge, the full plane field convergence for general $\beta$ still seems to be an open problem.

The microscopic version of Ward’s identity was introduced fairly recently in [3]. It is called Ward’s equation. See [3, Section 7.7] for the general case of $\beta$-ensembles. It is natural to ask how Ward’s equation fits into the present context. We hope to come back to this later on.