{"id":16090,"date":"2022-06-27T17:19:35","date_gmt":"2022-06-27T15:19:35","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=16090"},"modified":"2022-08-26T21:50:33","modified_gmt":"2022-08-26T19:50:33","slug":"unexpected-phenomena-for-equilibrium-measures","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2022\/06\/27\/unexpected-phenomena-for-equilibrium-measures\/","title":{"rendered":"Unexpected phenomena for equilibrium measures"},"content":{"rendered":"
\"Photo<\/a>
Marcel Riesz (1886 -1969)<\/figcaption><\/figure>\n

This post is about Riesz energy problems, a subject that I like to explore with Edward B. Saff<\/a> (Vanderbilt University, USA) and Robert S. Womersley<\/a> (UNSW Sydney, Australia).<\/p>\n

Riesz kernel.<\/strong> For $-2<s<d$, the Riesz $s$-kernel in $\\mathbb{R}^d$\u00a0 is $$
\nK_s:=\\begin{cases}
\n\\displaystyle\\frac{1}{s\\left|\\cdot\\right|^{s}} & \\text{if } s\\neq0\\\\[1em]
\n\\displaystyle-\\log\\left|\\cdot\\right| & \\text{if } s=0
\n\\end{cases}.
\n$$ We recover the Coulomb or Newton kernel when $s=d-2$. This definition of the $s$-kernel allows to pass from $K_s$ to $K_0$ by removing the $1\/s$ singularity at $s=0$, namely, for $x\\neq0$, $$-\\log|x|=\\lim_{\\underset{s\\neq0}{s\\to0}}\\frac{|x|^{-s}-1}{s-0}=\\lim_{\\underset {s\\neq0}{s\\to0}}\\Bigr(\\frac{1}{s|x|^s}-\\frac{1}{s}\\Bigr).$$<\/p>\n

Riesz energy.<\/strong> For $-2<s<d$, the Riesz energy of a probability measure $\\mu$ on $\\mathbb{R}^d$ is $$
\n\\mathrm{I}_s(\\mu):=\\iint K_s(x-y)\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)
\n=\\int(K_s*\\mu)\\mathrm{d}\\mu.
\n$$ The Riesz energy is strictly convex and lower semi-continuous for the weak convergence of probability measures with respect to continuous and bounded test functions. This convexity is related to the Bochner positivity of $K_s$, which is a nice observation from harmonic analysis.<\/p>\n

Equilibrium measure.<\/strong> The equilibrium measure on a ball $B_R:=\\{x\\in\\mathbb{R}^d:|x|\\leq R\\}$ is
\n$$
\n\\mu_{\\mathrm{eq}}
\n=\\arg\\min_{\\substack{\\mu\\\\\\mathrm{supp}(\\mu)\\subset B_R}}\\mathrm{I}_s(\\mu).
\n$$<\/p>\n

Riesz original problem (1938).<\/strong> Equilibrium measure on $B_R$ when $d\\geq2$ :
\n$$
\n\\mu_{\\mathrm{eq}}
\n=
\n\\begin{cases}
\n\\sigma_R & \\text{if $-2<s\\leq d-2$}\\\\[1em]
\n\\displaystyle\\frac{\\Gamma(1+\\frac{s}{2})}{R^s\\pi^{\\frac{d}{2}}\\Gamma(1+\\frac{s-d}{2})}
\n\\frac{\\mathbf{1}_{B_R}}{(R^2-|x|^2)^{\\frac{d-s}{2}}}\\mathrm{d}x &
\n\\text{if $0\\leq d-2<s<d$}
\n\\end{cases}
\n$$ where $\\sigma_R$ is the uniform distribution on the sphere $\\{x\\in\\mathbb{R}^d:|x|=R\\}$ of radius $R$.<\/p>\n

The proof relies on the following integral formula for the variational characterization : $$
\n\\int_{|y|\\leq R}
\n\\frac{|x-y|^{-s}}{(R^2-|y|^2)^{\\frac{d-s}{2}}}\\mathrm{d} y
\n=\\frac{\\pi^{\\frac{d}{2}+1}}{\\Gamma(\\frac{d}{2})\\sin(\\frac{\\pi}{2}(d-s))},\\,\\quad
\nx\\in B_R
\n$$ The Riesz proof of this integral formula involves in turn a Kelvin transform and a reduction to the planar case. It can be found in detail in the Appendix of the book by Landkof (1972), and also with even more details in our 2022 JMAA article. A generalization (and a new proof) was published in 2017 by Dyda, Kuznetsov, and Kwa\u015bnicki by using Fourier analysis.<\/p>\n

The result expresses a threshold phenomenon<\/strong> : the support condensates on a sphere when $s$ passes the critical value $d-2$ (Coulomb). Our main finding is that this Riesz problem admits a full space extension in which we replace the ball support constraint with an external field. We show that a new threshold phenomenon occurs, related to the strenght of the external field.<\/p>\n

External field equilibrium problem.<\/strong> The energy with external field $V$ on $\\mathbb{R}^d$ is defined by $$\\mathrm{I}(\\mu)=\\mathrm{I}_{s,V}(\\mu):=\\iint\\left[K_s(x-y)+V(x)+V(y)\\right]\\mathrm{d}\\mu(x)\\mathrm{d}\\mu(y)$$
\nand the associated equilibrium measure $$\\mu_{\\mathrm{eq}}=\\arg\\min_{\\mu}\\mathrm{I}(\\mu)$$ The Frostman or Euler-Lagrange variational characterization of $\\mu_{\\mathrm{eq}}$ reads $$K_s*\\mu+V
\n\\begin{cases}
\n=c& \\text{quasi-everywhere on }\\mathrm{supp}(\\mu)\\\\
\n\\geq c&\\text{quasi-everywhere outside }\\mathrm{supp}(\\mu)
\n\\end{cases}$$ Quasi-everywhere means except on a set that cannot carry a probability measure of finite energy. By taking $V=\\infty\\mathbf{1}_{B_R^c}$ we recover the Riesz problem on the ball mentioned previously.<\/p>\n

Coulomb case : $s=d-2$.<\/strong> The kernel $K_{d-2}$ is a Laplace fundamental solution :
\n$$
\n\\Delta K_{d-2}\\overset{\\mathcal{D}'}{=}-c_d\\delta_0,\\quad\\text{with}\\quad c_d=|\\mathbb{S}^{d-1}|.
\n$$Also, restricted to the interior of $\\mathrm{supp}(\\mu_{\\mathrm{eq}})$,
\n$$
\n\\mu_{\\mathrm{eq}}\\overset{\\mathcal{D}'}{=}\\frac{\\Delta V}{c_d}
\n$$In particular, if $V=\\left|\\cdot\\right|^\\alpha$, $\\alpha>0$, then
\n$$
\n\\mu_{\\mathrm{eq}}
\n=\\alpha(\\alpha+d-2)\\left|\\cdot\\right|^{\\alpha-2}\\mathbf{1}_{B_R}\\mathrm{d}x
\n\\quad\\text{with}\\quad R=\\bigr(\\frac{1}{\\alpha}\\bigr)^{\\frac{1}{d-2+\\alpha}}.$$ The proof relies crucially on the local nature of the Laplacian.<\/p>\n

At this point we observe that the formula $$\\Delta K_u=-c_{d,u}K_{u+2},\\quad c_{d,u}:=d-2-u$$ suggests to apply iteratively $\\Delta$ to reach the case $s=d-2n$ for an arbitrary positive integer $n$.<\/p>\n

Findings for the iterated Coulomb case $s=d-2n, n=1,2,3,\\ldots$.<\/strong> Then, restricted to the interior of $\\mu_{\\mathrm{eq}}$, in the sense of distributions,
\n$$
\n\\mu_{\\mathrm{eq}}
\n\\overset{\\mathcal{D}'}{=}
\n\\frac{\\Delta^{n}V}{c_dC_{d,n}},
\n\\quad\\text{where}\\quad
\nC_{d,n}:=(-1)^{n-1}\\prod_{k=0}^{n-2}c_{d,s+2k}=(-1)^{n-1}(2n-2)!!.
\n$$ In particular : if $s=d-4$ and $V=\\left|\\cdot\\right|^\\alpha$, $\\alpha\\geq2$, then $C_{d,2}<0$ while $\\Delta V=\\alpha(\\alpha+d-2)\\left|\\cdot\\right|^{\\alpha-2}\\geq0$ and thus $\\mu_{\\mathrm{eq}}$ is necessarily singular! Actually the case $s=d-4$ can be analyzed completely, and this analysis reveals the singularity when $\\alpha\\geq2$ as well as a threshold condensation to this singular support<\/strong> when $\\alpha$ reaches the critical value $2$.<\/p>\n

Findings when $s=d-4$.<\/strong> Suppose that $V=\\gamma\\left|\\cdot\\right|^\\alpha$, $\\gamma>0,\u00a0 \\alpha>0$.<\/p>\n