We present a general approach to study a class of random growth models in $n$-dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.

中文翻译：

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具有记忆的增长模型的平均原理和形状定理

我们提出了一种在 $n$ 维欧几里得空间中研究一类随机增长模型的一般方法。这些模型旨在捕捉基本的生长特征，这些特征预计会在最初在微观尺度上定义的几个经典自相互作用过程的介观水平上表现出来。它包括具有强强化的一次强化随机游走、原点激发的随机游走，以及其他少数几个，其中访问的顶点集有望形成“限制形状”。我们证明了导致这种形状定理的平均原理。可以根据关联马尔可夫链的不变度量来计算限制形状。