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Positive density of self-replicating initial conditions in large Game of Life grids

Establish the existence of a fixed δ>0 and threshold area A0 such that for all A≥A0, at least a δ-fraction of initial configurations on an A-area torus under Conway’s Game of Life exhibit non-trivial self-replication (according to a specified non-trivial necessary condition) at some point in their dynamics.

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Background

The authors advocate for probabilistic bounds on the emergence of self-replication from random initial conditions. They propose a concrete conjecture for Conway’s Game of Life that asserts the prevalence of self-replicating behavior on sufficiently large grids.

Proving such bounds would support the view that non-trivial self-replication may be generic under appropriate asymptotics.

References

We venture the following conjecture: Suppose we have a non-trivial necessary condition for non-trivial self-replication in finite area A. Consider the Game of Life CA with periodic boundary conditions on a square of area A. Then there exists a sufficiently large area A0 such that for all A \geq A0, there is a \delta > 0 such that at least a \delta-fraction of all initial conditions exhibit non-trivial self-replication at some point in their dynamics.

— Self-replication and Computational Universality (2510.08342 - Cotler et al., 9 Oct 2025) in Section 'Open problems and conjectures' (Probabilistic bounds for emergence of self-replication)