Roland Bauerschmidt, Gordon Slade

Research output: Contribution to journalArticlepeer-review


We provide an introductory account of a tricritical phase diagram, in the setting of a mean-field random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices N in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter g. A chemical potential ν controls the walk length. We determine the phase diagram in the (g, ν) plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order N) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.

Original languageEnglish (US)
Pages (from-to)167-204
Number of pages38
JournalProbability and Mathematical Physics
Issue number1
StatePublished - 2020


  • complete graph
  • mean field
  • phase transition
  • polymer model
  • theta point
  • tricritical point

ASJC Scopus subject areas

  • Statistics and Probability
  • Atomic and Molecular Physics, and Optics
  • Statistical and Nonlinear Physics


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