Heat Diffusion with Frozen Boundary

Laura Florescu, Shirshendu Ganguly, Yuval Peres, Joel Spencer

Research output: Contribution to journalArticlepeer-review


Consider “frozen random walk” on Z: n particles start at the origin. At any discrete time, the leftmost and rightmost (Formula presented.) particles are “frozen” and do not move. The rest of the particles in the “bulk” independently jump to the left and right uniformly. The goal of this note is to understand the limit of this process under scaling of mass and time. To this end we study the following deterministic mass splitting process: start with mass 1 at the origin. At each step the extreme quarter mass on each side is “frozen”. The remaining “free” mass in the center evolves according to the discrete heat equation. We establish diffusive behavior of this mass evolution and identify the scaling limit under the assumption of its existence. It is natural to expect the limit to be a truncated Gaussian. A naive guess for the truncation point might be the 1 / 4 quantile points on either side of the origin. We show that this is not the case and it is in fact determined by the evolution of the second moment of the mass distribution.

Original languageEnglish (US)
Pages (from-to)521-531
Number of pages11
JournalJournal of Statistical Physics
Issue number3
StatePublished - Nov 1 2015


  • Frozen mass
  • Moment evolution
  • Random walk
  • Truncated gaussian

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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