Structure of global minimizers of the interaction energy for mildly repulsive power-law potentials in the intermediate regime below the simplex threshold

The interaction energy $E(\mu)=\frac{1}{2}\iint W(x-y)\,d\mu(x)\,d\mu(y)$ over probability measures is the Lyapunov functional of the aggregation equation and models equilibrium states of collective behavior. For the power-law potential $W_{\alpha,\beta}(x)=\frac{|x|^{\alpha}}{\alpha}-\frac{|x|^{\beta}}{\beta}$ with attractive exponent $\alpha$ and mildly repulsive exponent $\beta>2$, Carrillo, Figalli and Patacchini (2017) proved that $d_{\infty}$-local minimizers have discrete support and global minimizers have finite support, with cardinality bounds. The open problem is stated explicitly by Davies, Lim and McCann (Comm. Math. Phys. 2023, Remark 1.7): determine the structure of the global minimizers in the intermediate regime $2<\beta<\alpha<\alpha_{\Delta}^{n}(\beta)$, lying below the threshold above which the unique minimizer is the uniform measure on the vertices of a regular unit-diameter $n$-simplex — the number of support points, their masses and geometric arrangement, uniqueness up to rigid motions, and the value of the threshold itself. In one dimension this contains the open questions of Kang, Kim, Lim and Seo (2021). The regime remains open in every dimension.

Background

Populations of interacting particles — cells in a tissue, animals in a swarm, or grains in a granular medium — are modeled at the continuum level by a Borel probability measure $\mu\in\mathcal{P}(\mathbb{R}^n)$ describing the mass distribution of agents, which interact pairwise through a potential $W$. The associated interaction energy is

$$E(\mu)=\frac{1}{2}\iint_{\mathbb{R}^n\times\mathbb{R}^n}W(x-y)\,d\mu(x)\,d\mu(y).$$

This functional decreases along solutions of the aggregation equation $\partial_t\mu=\nabla\cdot(\mu\nabla W*\mu)$, a continuity equation in which each agent moves down the gradient of the potential generated by the others; its minimizers are therefore the dynamically stable stationary states of collective-behavior models arising in biology, granular media, and economics. Potentials of interest are repulsive at short range and attractive at long range, and the canonical two-parameter family is the power-law potential

$$W_{\alpha,\beta}(x)=\frac{|x|^{\alpha}}{\alpha}-\frac{|x|^{\beta}}{\beta},\qquad\alpha>\beta,$$

whose attractive exponent $\alpha$ dominates at large distances (confining the mass) and whose repulsive exponent $\beta$ dominates near the origin (preventing collapse). Writing $W_{\alpha,\beta}(x)=w(|x|)$, the strength of the short-range repulsion is governed by $\beta$: for $\beta>2$ the potential is twice continuously differentiable at the origin with force $-w'(r)\sim r^{\beta-1}$ that vanishes faster than linearly — this is the mildly repulsive regime; $\beta=2$ is the borderline Hookean case, called the centrifugal line; $\beta<2$ is strongly repulsive, with a force that blows up at the origin.

How concentrated the minimizing measure is depends sharply on this exponent. Balagué, Carrillo, Laurent and Raoul (2013) tied the dimension of the support (the closed set on which a measure concentrates; its Hausdorff dimension measures its size allowing non-integer values) to the repulsion strength, showing that for mildly repulsive potentials every sufficiently smooth connected component of the support of a minimizer has Hausdorff dimension zero. Their argument, however, required a smoothness assumption on those components. Carrillo, Figalli and Patacchini (2017) removed that assumption: using the Euler–Lagrange conditions for local minimality with respect to the $\infty$-Wasserstein distance $d_{\infty}(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\sup_{(x,y)\in\mathrm{supp}\,\pi}|x-y|$ (the $L^{\infty}$ version of optimal-transport cost, where $\Pi(\mu,\nu)$ is the set of couplings, i.e. transport plans, between $\mu$ and $\nu$), they proved that for every radial $C^2$ mildly repulsive potential the support of any $d_{\infty}$-local minimizer consists of isolated points, and the support of any global minimizer is a finite set. For a global minimizer the convolution potential $V_{\mu}=W*\mu$ satisfies $V_{\mu}=2E(\mu)$ on the support and $V_{\mu}\ge 2E(\mu)$ everywhere, so once the support is known to be finite, minimality reduces to algebraic conditions on finitely many points and masses. The same paper bounded the number of support points: in one dimension $\mathrm{card}(\mathrm{supp}\,\mu)\le 2\lceil R/r\rceil+1$ with radii $R,r$ depending explicitly on the potential, while in dimensions $n\ge 2$ it obtained only a non-quantitative uniform bound under an additional $C^4$-type hypothesis — a gap the authors acknowledge explicitly, stating they obtain "a non-quantitative bound in any dimension, and a quantitative one in the one-dimensional case".

The quantitative gap left by Carrillo, Figalli and Patacchini defines the current research frontier: given that mildly repulsive minimizers are finite sums of Dirac masses (measures concentrated at single points), which finite measures are they? For the power-law family, all regimes except one are now classified. On the centrifugal line $\beta=2$ and in the strongly repulsive range $\beta<2$, Davies, Lim and McCann (2022–2023) and Frank and Matzke (2025) proved that minimizers are uniform measures on spheres, except in one dimension for $\alpha\in(2,3)$ where Frank (2022) showed the minimizer is absolutely continuous with an unbounded density. In the mildly repulsive regime $2<\beta<\alpha$, Davies, Lim and McCann (2023) identified a transition threshold $\alpha_{\Delta}^{n}(\beta)$ — a decreasing homeomorphism of $[2,4]$ onto itself with $\alpha_{\Delta}^{n}(\beta)\le\max\{\beta,4\}$ and $\alpha_{\Delta}^{n}(2)=4$ for $n\ge 2$ — such that for $\alpha\ge\alpha_{\Delta}^{n}(\beta)$ the unique global minimizer up to rigid motions is the uniform measure on the $n+1$ vertices of a regular simplex of unit diameter (a regular simplex being the convex hull of points with all pairwise distances equal: an equilateral triangle in the plane, a regular tetrahedron in space). Below that threshold nothing is known beyond the finiteness theorem and cardinality bound of Carrillo, Figalli and Patacchini: Davies, Lim and McCann state as an open problem to determine the structure of minimizers for $2<\beta<\alpha<\alpha_{\Delta}^{n}(\beta)$, remarking that "little else is known about this subregime". In one dimension, where the unit simplex is the two-point measure $\frac{1}{2}(\delta_{0}+\delta_{1})$, Kang, Kim, Lim and Seo (2021) proved that this two-point measure is the unique local minimizer once the attraction exponent exceeds a threshold $p^{*}(\beta)$, and exhibited the exponents $(\alpha,\beta)=(2.5,2.1)$ at which a three-point measure with unequal masses has strictly lower energy — so the intermediate regime genuinely contains nontrivial minimizers whose structure is unknown.

Problem Statement

For the interaction energy

$$E_{\alpha,\beta}(\mu)=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\left(\frac{|x-y|^{\alpha}}{\alpha}-\frac{|x-y|^{\beta}}{\beta}\right)d\mu(x)\,d\mu(y),\qquad\mu\in\mathcal{P}(\mathbb{R}^{n}),$$

where $\mathcal{P}(\mathbb{R}^{n})$ denotes the set of Borel probability measures on $\mathbb{R}^{n}$ and the exponents are mildly repulsive, $2<\beta<\alpha$ (attractive exponent $\alpha$, repulsive exponent $\beta$), determine the global minimizers on $\mathcal{P}(\mathbb{R}^{n})$ in the intermediate regime left open by the known classifications: for each dimension $n\ge 1$ and each pair $2<\beta<\alpha$ for which the uniform measure on the vertices of a regular unit-diameter $n$-simplex is not the global minimizer (equivalently, $2<\beta<\alpha<\alpha_{\Delta}^{n}(\beta)$, with the threshold understood in the modified one-dimensional sense of Davies, Lim and McCann when $n=1$), identify the minimizing measure(s): the number of support points, their masses, their geometric arrangement, and whether the minimizer is unique up to rigid motions (translations and rotations).

The problem has three linked components within this single classification objective: (i) sharp quantitative information on the number of support points in dimensions $n\ge 2$, where the source paper's quantitative estimate is one-dimensional only; (ii) the determination of the transition threshold $\alpha_{\Delta}^{n}(\beta)$ itself on the range $\beta\in(2,4)$, in particular whether it coincides with the lower bound obtained from the pointwise Euler–Lagrange condition; and (iii) in dimension $n=1$, where the unit simplex is the two-point measure $\frac{1}{2}(\delta_{0}+\delta_{1})$ and Kang, Kim, Lim and Seo write $(p,q)$ for the exponent pair $(\alpha,\beta)$: what is a global minimizer in the exponent range of their Proposition 2, in which $\frac{1}{2}(\delta_{0}+\delta_{1})$ fails to be even a local minimizer (their Q1); is $\frac{1}{2}(\delta_{0}+\delta_{1})$ the global minimizer whenever $\alpha>\beta>q^{*}$, where $q^{*}$ is the unique root of $q\log 2=2^{q-1}-1$ (their Q2); does the failure threshold $p_{*}(\beta)$ of their Proposition 2 coincide with the uniqueness threshold $p^{*}(\beta)$ of their Theorem 1 — the attraction exponent above which $\frac{1}{2}(\delta_{0}+\delta_{1})$ is the unique local minimizer up to translation — on $2\le\beta<q^{*}$ (their Q3); and does $p^{*}(\beta)=\beta$ hold for all sufficiently large $\beta$ (their Conjecture). All three components are required for the classification, and the four questions listed in (iii) are independent of one another: resolving any one of them settles only that question.

References

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