Single-crystal neutron scattering and magnetic susceptibility measurements of the dilute magnetic topological insulator (${\mathrm{Sn}}_{0.9}{\mathrm{Mn}}_{0.1})\mathrm{Te}$ demonstrate the emergence of long-range ferromagnetic (FM) order below $T_{\mathrm{C}}=12\pm 1\mathrm{K}$. Through inelastic and quasielastic neutron scattering analyses, we uncover a complex interplay of competing magnetic interactions, including nearest-neighbor (NN) FM Mn-Mn exchange, second-NN antiferromagnetic (AFM) Mn-Te-Mn superexchange, and progressively weaker long-range interactions extending to at least the eighth NN. The AFM interaction mediated by the linear Mn-Te-Mn bond ($J_2\sim 0.5\mathrm{meV}$) is the strongest and leads to Mn-Mn dimer singlet states that do not contribute to FM ordering. Instead, we identify the long-range seventh-NN FM interaction $J_7$ as the most effective channel for stabilizing the FM order within a minimal model consistent with the data. Despite its relatively weak strength, its large coordination number ($Z=48$) enables percolation of exchange bonds, effectively stabilizing the FM state. Reverse Monte Carlo and atomistic spin dynamics simulations of the neutron scattering data are consistent with the observed FM state of dilute (${\mathrm{Sn}}_{0.9}{\mathrm{Mn}}_{0.1})\mathrm{Te}$. These findings underscore the critical role of long-range percolative interactions in stabilizing FM order in dilute magnetic topological systems. Beyond topological materials, the results are broadly relevant to magnetism in semiconductors and dilute magnetic alloys, where competing exchange pathways and percolative effects similarly govern the emergence of collective magnetic order.

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• Exchange interaction
• Ferromagnetism
• Percolation phase transition
• Spin dynamics
• Dilute magnetic semiconductors
• Semiconductor compounds
• Topological materials
• Magnetization measurements
• Monte Carlo methods
• Neutron scattering