Optimal correlation scaling for singular interactions

Determine whether, for the Langevin dynamics of N exchangeable Brownian particles with singular pairwise interactions K(x,y) = −∇W(x−y) belonging to L^2((T^d)^2)^d and with W satisfying the stated exponential integrability conditions, the m-particle nonlinear correlation functions G_{N,m} defined via the cluster expansion admit bounds sharper than the established L^2(ω_{β_*}^{⊗ m}) estimates G_{N,m} = O(N^{−m/2}) on a fixed time interval; in particular, ascertain whether the BBGKY-suggested optimal scaling G_{N,m} = O(N^{1−m}) can be rigorously established in this singular interaction setting.

Background

The paper studies correlation functions (Ursell functions) for Brownian particle systems with pairwise interactions in the mean-field scaling, focusing on potentially unbounded (singular) interaction kernels. For smooth or bounded interactions, formal BBGKY hierarchy arguments suggest the optimal scaling G_{N,m} = O(N{1−m}).

To handle singular interactions, the authors develop a framework of linearized correlation functions and obtain robust weighted L2 bounds for nonlinear correlations, showing G_{N,m} = O(N{−m/2}) and a weak convergence result N{m/2} G_{N,m} ⇀ 0 on a fixed time interval. These results extend control of correlations beyond the bounded interaction regime but are suboptimal compared to the expected scaling.

The authors explicitly note that it is currently unknown whether stronger correlation estimates are possible in the singular setting they consider, leaving open the question of achieving the optimal mean-field scaling or any improvement beyond N{−m/2} in weighted L2 norms.

References

Our estimates do not match the expected scaling~eq:scaling-GNm: we only show $G_{N,m}=O(N{-m/2})$ in $L2$, cf.~eq:L2bnd below, together with the weak convergence $N{m/2}G_{N,m}\rightharpoonup0$, cf.~eq:wL2bnd. We do not know whether stronger estimates can be expected to hold in the present singular setting.

eq:scaling-GNm:

GN,m=O(N1−m).G_{N,m}=O(N^{1-m}).

eq:L2bnd:

(∫m∣GN,m(t)∣2ωβ∗⊗m)12 ≤ CmN−m2,\Big(\int_{^m}|G_{N,m}(t)|^2\omega_{\beta_*}^{\otimes m}\Big)^\frac12\,\le\, C_mN^{-\frac m2},

eq:wL2bnd:

$N^{\frac m2}G_{N,m}\overset*\rightharpoonup0,\qquad\text{in $L^\infty(0,T_*;L^2(\omega_{\beta_*}^{\otimes m}))$}. $

— Correlation estimates for Brownian particles with singular interactions (2510.01507 - Duerinckx et al., 1 Oct 2025) in Section 1.2 (Main results), paragraph preceding Theorem 1.1