Percolation problems appear in a large variety of different contexts ranging
from the design of composite materials to vaccination strategies on community
networks. The key observable for many applications is the percolation
threshold. Unlike the universal critical exponents, the percolation threshold
depends explicitly on the specific system properties. As a consequence,
theoretical approaches to the percolation threshold are rare and generally
tailored to the specific application.
Yet, any percolating cluster forms a discrete network the emergence of which
can be cast as a graph problem and analyzed using branching processes. We
propose a general mapping of any kind of percolation problem onto a branching
process which provides rigorous lower bounds of the percolation threshold.
These bounds progressively tighten as we incorporate more information into the
theory. We showcase our approach for different continuum problems finding
accurate predictions with almost no effort. Our approach is based on first
principles and does not require fitting parameters. As such it offers an
important theoretical reference in a field that is dominated by simulation
studies and heuristic fit functions.Comment: 8 pages, 5 figure