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Quantum gravitational effects become significant at a cut-off species scalethat can be much lower than the Planck scale whenever we get a parametricallylarge number of fields becoming light. This is expected to occur at anyperturbative limit of an effective field theory coupled to gravity, orequivalently, at any infinite distance limit in the field space of the quantumgravity completion. In this note, we present a universal pattern that links theasymptotic variation rates in field space of the quantum gravity cut-off$\Lambda_{\text{sp}}$ and the characteristic mass of the lightest tower ofstates $m$: $\frac{\vec\nabla m}{m} \cdot\frac{\vec\nabla \Lambda_{\rm sp}}{\Lambda_{\rm sp}}=\frac1{d-2}$, where $d$ is the spacetime dimension. Thisrestriction can be used to make more precise several Swampland criteria thatconstrain the effective field theories that can be consistently coupled toquantum gravity.

A Universal Pattern in Quantum Gravity at Infinite Distance