{
  "id": "math/0603109",
  "version": "v1",
  "published": "2006-03-04T15:36:59.000Z",
  "updated": "2006-03-04T15:36:59.000Z",
  "title": "Threshold $theta geq 2$ contact processes on homogeneous trees",
  "authors": [
    "Luiz Renato Fontes",
    "Roberto H. Schonmann"
  ],
  "comment": "27 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the threshold $theta geq 2$ contact process on a homogeneous tree $T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and started from a product measure with density $p$. The corresponding mean-field model displays a discontinuous transition at a critical point $lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}(kappa,theta)$ it survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty} p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process on $T_b$ has a qualitatively similar behavior when $lambda$ is small, including the behavior at and close to the critical point $lambda_c(T_b,theta)$. In contrast, for large $lambda$ the behavior of the process on $T_b$ is qualitatively distinct from that of the mean-field model in that the critical density has $p_c(T_b,theta,infty) := lim_{lambda to infty} p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$, $lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty} b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty} b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-03-04T15:36:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "homogeneous tree",
      "contact processes",
      "corresponding mean-field model displays",
      "critical point",
      "product measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3109F"
    }
  }
}