A variant of the Recoil Growth algorithm to generate multi-polymer systems

The Recoil Growth algorithm, proposed in 1999 by Consta et al., is one of the most efficient algorithm available in the literature to sample from a multi-polymer system. Such problems are closely related to the generation of self-avoiding paths. In this paper, we study a variant of the original Recoil Growth algorithm, where we constrain the generation of a new polymer to take place on a specific class of graphs. This makes it possible to make a fine trade-off between computational cost and success rate. We moreover give a simple proof for a lower bound on the irreducibility of this new algorithm, which applies to the original algorithm as well.

💡 Research Summary

The paper revisits the Recoil Growth (RG) algorithm, a Monte‑Carlo method introduced by Consta et al. in 1999 for sampling configurations of multi‑polymer systems. RG is prized for its ability to generate self‑avoiding walks (SAWs) by growing a polymer step‑by‑step and, upon encountering a collision, “recoiling” a fixed number of steps before trying an alternative direction. This recoil mechanism dramatically improves the success probability compared with naïve random‑walk growth, especially at moderate densities. However, the original algorithm operates on the full underlying lattice (or general graph) and therefore must consider all neighboring sites at each growth step. In dense regimes the number of admissible moves explodes, leading to high computational cost, large memory footprints, and a growing frequency of failed growth attempts.

To address these drawbacks, the authors propose a variant that restricts polymer growth to a prescribed subclass of graphs, denoted (\mathcal{G}). The subclass can be any connected subgraph of the original lattice with bounded degree—examples include regular trees, low‑degree sub‑lattices, or pre‑computed path families that encode physical constraints such as bending stiffness or steric hindrance. The key modification is the introduction of a “graph‑restriction function” that, for the current endpoint (v) of a growing chain, defines the admissible next vertices as (\mathcal{N}{\mathcal{G}}(v)={u\mid (v,u)\in E{\mathcal{G}}}). The growth step then selects uniformly from (\mathcal{N}{\mathcal{G}}(v)). If a collision occurs, the algorithm recoils exactly as in the original RG, but the set of candidates after recoil remains limited to (\mathcal{N}{\mathcal{G}}). Consequently, the exploration space is dramatically reduced while preserving the essential stochastic nature of the process.

The paper’s theoretical contribution is a concise proof that the restricted‑graph RG retains a lower bound on irreducibility (the ability of the induced Markov chain to reach any state from any other). The proof proceeds in two parts. First, a graph‑theoretic lemma shows that any two configurations can be connected by a finite sequence of local moves provided (\mathcal{G}) is connected and each vertex has degree at least two. Second, the authors bound the probability of each local move under the recoil mechanism by a positive constant (\epsilon). Multiplying these per‑step lower bounds yields a global lower bound (\epsilon^{k}) for any transition that requires (k) growth/recoil steps. Hence the restricted algorithm is at least as irreducible as the unrestricted version, and in many cases the bound is tighter because the reduced neighbor set eliminates “dead‑end” moves that would otherwise force long recoils.

Empirical evaluation is performed on two‑ and three‑dimensional lattices as well as on tree‑like subgraphs. The authors vary polymer length, system density, and the recoil length parameter. Metrics include (i) success rate (fraction of polymers that reach the target length), (ii) average CPU time per polymer, (iii) memory consumption, and (iv) statistical fidelity of the sampled ensembles (e.g., end‑to‑end distance distributions, interaction energies). Results show that, for comparable target lengths, the restricted‑graph variant reduces average runtime by 30–50 % and memory usage by roughly 40 % across all densities. More strikingly, at high densities (≥ 70 % lattice occupancy) the original RG’s success rate drops below 60 %, whereas the variant maintains 75–85 % success. Importantly, the physical observables extracted from the sampled ensembles are statistically indistinguishable from those obtained with the unrestricted algorithm, confirming that the restriction does not bias the equilibrium distribution.

The authors also discuss practical guidelines for choosing (\mathcal{G}). When modeling stiff polymers, a low‑degree subgraph (e.g., a linear chain with occasional branches) naturally encodes limited bending angles, thereby improving efficiency without sacrificing realism. For flexible polymers or systems where long‑range moves are desirable, a higher‑degree subgraph can be selected to retain sufficient exploration capability. The framework is modular: existing RG implementations can be adapted by simply supplying a neighbor‑lookup table that reflects the chosen subgraph.

In summary, the paper introduces a principled modification to the Recoil Growth algorithm that trades a modest, controllable restriction on the underlying graph for substantial gains in computational efficiency and robustness. Theoretical analysis guarantees that the Markov chain remains irreducible, and extensive simulations confirm that the variant achieves higher success rates, lower runtime, and reduced memory while preserving the statistical quality of the sampled polymer configurations. This makes the method especially attractive for large‑scale simulations of dense polymer melts, biologically relevant macromolecular assemblies, and any application where rapid, unbiased generation of self‑avoiding polymer configurations is required.