A Verlet list (named after Loup Verlet) is a data structure in molecular dynamics simulations to efficiently maintain a list of all particles within a given cut-off distance of each other.
This method may easily be applied to Monte Carlo simulations. For short-range interactions, a cut-off radius is typically used, beyond which particle interactions are considered "close enough" to zero to be safely ignored. For each particle, a Verlet list is constructed that lists all other particles within the potential cut-off distance, plus some extra distance so that the list may be used for several consecutive Monte Carlo "sweeps" (set of Monte Carlo steps or moves) before being updated. If we wish to use the same Verlet list formula_1 times before updating, then the cut-off distance for inclusion in the Verlet list should be formula_2, where formula_3 is the cut-off distance of the potential, and formula_4 is the maximum Monte Carlo step (move) of a single particle. Thus, we will spend of order formula_5 time to compute the Verlet lists (formula_6 is the total number of particles), but are rewarded with formula_1 Monte Carlo "sweeps" of order formula_8 instead of formula_9. By optimizing our choice of formula_1 it can be shown that Verlet lists allow converting the formula_11 problem of Monte Carlo sweeps to an formula_12 problem.
Using cell lists to identify the nearest neighbors in formula_13 further reduces the computational cost.
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