Structure of the stationary state of the asymmetric target process

Abstract

Abstract. We introduce a novel migration process, the target process. This process is dual to the zero-range process (ZRP) in the sense that, while for the ZRP the rate of transfer of a particle only depends on the occupation of the departure site, it only depends on the occupation of the arrival site for the target process. More precisely, duality associates to a given ZRP a unique target process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of a bias, both processes have the same stationary-state product measure. In this work we focus our interest on the situation where the latter measure exhibits a continuous condensation transition at some finite critical density ρ c , irrespective of the dimensionality. The novelty comes from the case of asymmetric dynamics, where the target process has a nontrivial fluctuating stationary state, whose characteristics depend on the dimensionality. In one dimension, the system remains homogeneous at any finite density. An alternating scenario however prevails in the high-density regime: typical configurations consist of long alternating sequences of highly occupied and less occupied sites. The local density of the latter is equal to ρ c and their occupation distribution is critical. In dimension two and above, the asymmetric target process exhibits a phase transition at a threshold density ρ 0 much larger than ρ c . The system is homogeneous at any density below ρ 0 , whereas for higher densities it exhibits an extended condensate elongated along the direction of the mean current, on top of a critical background with density ρ c .

Condensation in Stochastic Particle Systems with Stationary Product Measures

“…The dynamics of the model is clearly not well defined in the thermodynamic limit at least for supercritical densities, while this is expected to be the case for the ZRP with bounded rates (40), even though this is not proven to our knowledge. Note that in contrast to the strikingly different dynamics, which is encoded in the different asymptotic behaviour of the jump rates, the static stationary behaviour for both models is in fact identical.…”

“…The recently introduced explosive condensation model [11], also discussed in more detail in Section 3.2, has the same rich structure in terms of stationary product measures and poses interesting questions for future work with respect to the dynamics. For the other two models mentioned in Section 2.1, the target process [40] has only a restricted set of stationary product measures and further progress is very challenging, and the inclusion process exhibits homogeneous condensation only in a particular scaling limit with system size dependent parameters [18], which is still under investigation [57].…”

“…Case 2 is a straightforward extension to a proof for inclusion processes given in [18] including one of the authors, which is based on a classical result on the exclusion process [2, Theorem VIII.2.1]. Cases 2 and 3 have also been discussed in [40] in the context of the target process. Another recent model covered by case 3 is the explosive condensation model studied in [11] for totally asymmetric dynamics on a one-dimensional torus, and the theorem also holds for other geometries.…”

Invariant measures of mass migration processes

“…for a nonnegative function g 1 ·,· on N × N, non-decreasing (non-increasing) in its first (second) coordinate. In the works of Godrèche et al [15,20,21], arbitrary (that is, without monotonicity properties) nonnegative g 1 ·,· on N × N are considered in (18) and the process is then called the dynamic urn model or the migration process. However, as per usual, we keep the denomination MP also for those cases; we denote by mass migration processes (MMP) the dynamics extended to multiple jumps.…”

“…Conservative systems such as exclusion or zero-range processes possess a oneparameter family of translation invariant and invariant product probability measures, where the parameter represents the average particle density per site [26,19,1]. Under conditions on the rates, this is also the case for misanthropes processes, which include the more recently studied target processes [8,15,21]. All these dynamics (we call them single-jump models) consist in individual jumps of particles between sites, with rates which are the product of two terms: a transition probability giving the direction of the jump, and a function depending on the occupation numbers at the departure and/or arrival sites of the jump.…”