Quenched localisation in the Bouchaud trap model with regularly varying traps

Localisation in the Bouchaud-Anderson Model

It is well-known that both branching random walk models and trap models can exhibit intermittency and localisation phenomena; the prototypical examples being the parabolic Anderson and Bouchaud trap models respectively. Our aim is to investigate how these localisation phenomena interact. To do so, we study a hybrid model combining the dynamics of both the parabolic Anderson and the Bouchaud trap models; more precisely, we consider a variant of the parabolic Anderson model in which the underlying random walk driving the model is replaced with the Bouchaud trap model. In this initial study, we c…

具有缓变陷阱的Bouchaud陷阱模型中的两点局部化

Two-site localisation in the Bouchaud trap model with slowly varying traps

我们考虑整数上的布绍陷阱模型，陷阱分布在无穷远处有一个缓慢变化的尾部。我们证明，该模型最终定位在两个网站上以压倒性的概率。这是一个更强的形式的本地化比以前已经建立在文献中的布绍陷阱模型的整数的情况下，定期变化的陷阱。这个结果的基础是这样一个事实，即i.i.d.具有缓变尾部的随机变量是由极大项渐近控制的。

Quenched localisation in the Bouchaud trap model with slowly varying traps

We consider the quenched localisation of the Bouchaud trap model on the positive integers in the case that the trap distribution has a slowly varying tail at infinity. Our main result is that for each $N \in \{2, 3, \ldots\}$ there exists a slowly varying tail such that quenched localisation occurs on exactly $N$ sites. As far as we are aware, this is the first example of a model in which the exact number of localisation sites are able to be `tuned' according to the model parameters. Key intuition for this result is provided by an observation about the sum-max ratio for sequences of independen…

The Bouchaud-Anderson model with double-exponential potential

The Bouchaud-Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneou…

具有缓变陷阱的Bouchaud陷阱模型的泛函极限定理

Functional limit theorems for the Bouchaud trap model with slowly varying traps

我们考虑整数上的布绍陷阱模型，陷阱分布在无穷远处有一个缓慢变化的尾部。我们的主要结果是一个功能的极限定理的退火法律下的模型，类似的功能极限定理以前建立在文献中的情况下，可积或定期变化的陷阱分布。反映的事实，时钟过程是占主导地位的极限的贡献，从最深的访问陷阱，该模型的极限过程是一个空间从属的布朗运动，其相关的时钟过程是一个极值过程。