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We study independent long-range percolation on $\mathbb{Z}^d$ where thevertices $x$ and $y$ are connected with probability$1-e^{-\beta\|x-y\|^{-d-\alpha}}$ for $\alpha > 0$. Provided the criticalexponents $\delta$ and $2-\eta$ defined by $\delta = \lim_{n\to \infty}\frac{-\log(n)}{\log\left(\mathbb{P}_{\beta_c}\left(|K_0|\geq n\right)\right)}$and $2-\eta = \lim_{x \to \infty}\frac{\log\left(\mathbb{P}_{\beta_c}\left(0\leftrightarrowx\right)\right)}{\log(\|x\|)} + d$ exist, where $K_0$ is the cluster containingthe origin, we show that  \begin{equation*}  \delta \geq \frac{d+(\alpha\wedge 1)}{d-(\alpha\wedge 1)} \ \text{ and } \2-\eta \geq \alpha \wedge 1 \text.  \end{equation*}  The lower bound on $\delta$ is believed to be sharp for $d = 1, \alpha \in\left[\frac{1}{3},1\right)$ and for $d = 2, \alpha \in\left[\frac{2}{3},1\right]$, whereas the lower bound on $2-\eta$ is sharp for$d=1, \alpha \in (0,1)$, and for $\alpha \in \left(0,1\right]$ for $d>1$, andis not believed to be sharp otherwise. Our main tool is a connection betweenthe critical exponents and the isoperimetry of cubes inside $\mathbb{Z}^d$.

Isoperimetric lower bounds for critical exponents for long-range percolation