Condensation of SIP Particles and Sticky Brownian Motion

Abstract

We study the symmetric inclusion process (SIP) in the condensation regime. We obtain an explicit scaling for the variance of the density field in this regime, when initially started from a homogeneous product measure. This provides relevant new information on the coarsening dynamics of condensing interacting particle systems on the infinite lattice. We obtain our result by proving convergence to sticky Brownian motion for the difference of positions of two SIP particles in the sense of Mosco convergence of Dirichlet forms. Our approach implies the convergence of the probabilities of two SIP particles to be together at time t. This, combined with self-duality, allows us to obtain the explicit scaling for the variance of the fluctuation field.

Correction: Second time scale of the metastability of reversible inclusion processes

“…Originally in [7,], the mode of convergence in part (1) was alternatively chosen as the convergence of trace processes in the Skorokhod topology. Combining this result with an additional technicality presented in [33,Proposition 2.1], we may easily deduce the convergence of finite-dimensional marginal distributions as presented in Theorem 2.6- (1).…”

“…where the inequality holds by the uniform boundedness proved in part (1). By the strong Markov property and (4.21), since T L ≥ τ L−ℓ , the last term is bounded by…”

“…where E L • is the expectation corresponding to the law of {X L t } t≥0 . Thus, to conclude the proof of part (1), it remains to prove that the solutions are uniformly bounded.…”

“…when the factor d N controlling the diffusive behavior of the system vanishes in a certain manner as the particle number N tends to infinity (see (2.4) and (2.5)). The results of condensation were further extended to inclusion processes under various conditions on d N and with different geometries of the underlying space [1,24]. We remark that non-condensing regimes were also thoroughly investigated in the community during the past few years, regarding the topics of duality [12][13][14], local equilibrium properties [28,42], non-equilibrium limit theorems [17,18], etc.…”