Leading order asymptotics for non-local energies and the Read-Shockley law

We study an energy minimization problem ${\sum }_{i\mathrm{\ne }j}W(z_i-z_j)$ for $N$ points $\{z_1,\dots ,z_N\}$ with applications in dislocation theory. The $N$ points lie in the two-dimensional domain $\mathbb{R}\times [-\pi ,\pi ]$, %who are trying to minimize their interaction energy where where the kernel $W$ is derived from the Volterra potential $V(x,y)=\frac{x^2}{x^2+y^2}-\frac{1}{2}\log (x^2+y^2)$. We prove that the minimum energy is given by $-N\log N+\mathcal{O}(N)$. This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals.