If $\rho=Q$, then $$\label{11july13e8}
    \sum_i\bigg (\nabla (\log h_i)(p_k)+8\pi  \nabla_1 \gamma(p_k,p_k) \bigg )\rho_{i}=O(\epsilon_k^2\log \epsilon_k^{-1}).$$ where $\nabla_1$ denotes the derivative with respect to the first variable, and $\gamma(x,y)$ stands for the regular part of the Green’s function.

Our second result is about the decay rate of $\Lambda_I(\rho^k)$. To state the result, we need to define the following quantity $D_i$ ($i\in I=\{1,...,n\}$): $$\label{leadingD}
D_i=\lim_{\delta_0\to 0}\big (\delta_0^{2-m}-\frac{m-2}{2\pi}
\int_{M\setminus B(p,\delta_0)}\frac{h_i(x)}{h_i(p)}e^{2\pi m (G(x,p)-\gamma(p,p))}dV_g \big )$$ provided that $m<4$. The limit is well defined if $m<4$, see section 6.

\[rhoto1mle4\] Suppose $\rho^k\to \rho\in \Gamma_1$ and $\rho\neq Q$, if all $\rho_i^k-\rho_i$ have the same sign, then $$\Lambda_I(\rho^k)
=8\pi^2\sum_{i\in I_1}(e^{c_i}D_i+o(1))\epsilon_k^{m-2},$$ where $I_1$ is a subset of $I$ where $m_i=m$ for all $i\in I_1$, $c_i$ is a constant determined in (\[12jun29e1\]). $o(1)\to 0$ as $k\to \infty$.

If $M$ is a flat torus with fundamental cell domain $\Omega\subset \mathbb R^2$, then $D_i$ can be written as $$D_i=\frac{m-2}{2\pi }\bigg (
\int_{\mathbb R^2\setminus \Omega} \frac{1}{|x-p|^m}dx-\lim_{\delta_0\to 0}\int_{\Omega\setminus B_{\delta_0}}
\frac{H_i(x,p)}{|x-p|^m}dx\bigg )$$ where $$H_i(x,p)=\frac{h_i(x)}{h_i(p)}e^{2\pi m(\gamma(x,p)-\gamma(p,p))}-1, \quad i\in I.$$ See [@ccl] and [@clwang] for related discussions.

The assumption that all $\rho_i^k-\rho_i$ have the same sign seems due to some technical difficulties. When $n=2$ and $\rho\neq Q$, this assumption is not needed for both Theorem \[locations\] and Theorem 