2001/11/30 by S. J. O'Donoghue, S. J. O’Donoghue, A. J. Bray
Physics and Astronomy · #Complex Network Analysis Techniques #Opinion Dynamics and Social Influence #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.65.051114
published as Phys. Rev. E 65, 051114 (2002) · Expanded introduction with more discussion of related work
arxiv created 2002/03/06 · openalex publication_date 2002/05/20 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The dynamics of the one-dimensional q-state Potts model, in the zero-temperature limit, can be formulated through the motion of random walkers which either annihilate (A+\stackrel\ensuremath→A\ensuremath∅) or coalesce (A+\stackrel\ensuremath→AA) with a q-dependent probability. We consider all of the walkers in this model to be mutually infectious. Whenever two walkers meet, they experience mutual contamination. Walkers which avoid an encounter with another random walker up to time t remain uninfected. The fraction of uninfected walkers is known to obey a power-law decay U(t)\ensuremath∼t^\ensuremath-\ensuremathφ(q), with a nontrivial exponent \ensuremathφ(q) [C. Monthus, Phys. Rev. E 54, 4844 (1996); S. N. Majumdar and S. J. Cornell, ibid. 57, 3757 (1998)]. We probe the numerical values of \ensuremathφ(q) to a higher degree of accuracy than previous simulations and relate the exponent \ensuremathφ(q) to the persistence exponent \ensuremathθ(q) [B. Derrida, V. Hakim, and V. Pasquier, Phys. Rev. Lett. 75, 751 (1995)], through the relation \ensuremathφ(q)=\ensuremathγ(q)\ensuremathθ(q) where \ensuremathγ is an exponent introduced in [S. J. O'Donoghue and A. J. Bray, preceding paper, Phys. Rev. E 65, XXXX (2002)]. Our study is extended to include the coupled diffusion-limited reaction A+\stackrel\ensuremath→AB, B+\stackrel\ensuremath→BA in one dimension with equal initial densities of A and B particles. We find that the density of walkers decays in this model as \ensuremathρ(t)\ensuremath∼t^\ensuremath-1/2. The fraction of sites unvisited by either an A or a B particle is found to obey a power law, P(t)\ensuremath∼t^\ensuremath-\ensuremathθ with \ensuremathθ\ensuremath≃1.33. We discuss these exponents within the context of the q-state Potts model and present numerical evidence that the fraction of walkers which remain uninfected decays as U(t)\ensuremath∼t^\ensuremath-\ensuremathφ, where \ensuremathφ\ensuremath≃1.13 when infection occurs between like particles only, and \ensuremathφ\ensuremath≃1.93 when we also include cross-species contamination. We find that the relation between \ensuremathφ and \ensuremathθ in this model can also be characterized by an exponent \ensuremathγ, where similarly, \ensuremathφ=\ensuremathγ\ensuremathθ.