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We study stable solutions to the fractional Allen-Cahn equation \linebreak$(-\Delta)^{s/2} u = u-u^3$, $|u|<1$ in $\mathbb{R}^n$. For every $s\in (0,1)$and dimension $n\geq 2$, we establish sharp energy estimates, densityestimates, and the convergence of blow-downs to stable nonlocal $s$-minimalcones. As a consequence, we obtain a new classification result: if for somepair $(n,s)$, with $n\ge 3$, hyperplanes are the only stable nonlocal$s$-minimal cones in $\mathbb{R}^n\setminus\{0\}$, then every stable solutionto the fractional Allen-Cahn equation in $\mathbb{R}^n$ is 1D, namely, itslevel sets are parallel hyperplanes.  Combining this result with the classification of stable $s$-minimal cones in$\mathbb{R}^3\setminus\{0\}$ for $s\sim 1$ obtained by the authors in a recentpaper, we give positive answers to the "stability conjecture" in $\mathbb{R}^3$and to the "De Giorgi conjecture" in $\mathbb{R}^4$ for the fractionalAllen-Cahn equation when the order $s\in (0,1)$ of the operator is sufficientlyclose to $1$.

Stable solutions to the fractional Allen-Cahn equation in the nonlocal perimeter regime