We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear.

中文翻译：

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具有离散性引起的转变的系统的平稳分布

我们对一些表现出离散诱导转变现象的自催化反应网络进行了理论分析。我们讨论的网络家族包括著名的 Togashi 和 Kaneko 模型。我们证明了族中模型的正递归、所有矩的有限性和几何遍历性。对于某些参数值，我们找到了平稳分布的解析表达式，并讨论了体积缩放对链的平稳行为的影响。我们找到了离散引起的转变消失的体积的确切临界值。