A proof of convergence for Ant algorithms

A proof of convergence for Ant algorithms is developed. Ant algorithms were modeled as branching random processes: the branching random walk and branching Wiener process to derive rates of birth and death of ant paths. Substitution is then carried out in birth-death processes, which proves that a stable distribution is surely reached. This indicates that Ant algorithms converge with probability one. This analogy models Ant algorithms complexity parameters such as the number of cycles, the degrees of freedom of problem and the number of ants.