We introduce random walks in a sparse random environment on Z and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly distributed over the integer lattice. Interestingly, in the critical (recurrent) regime, our model generalizes Sinai's scaling of (log n) 2 for the location of the random walk after n steps to (log n) α , where α > 0 is a parameter determined by the distribution of the distance between two successive impurities. Similar scaling factors have appeared in the literature in different contexts and have been discussed in [29] and [31]. MSC2010: primary 60K37; secondary 60F05.

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Dementia is a degenerative disease that is increasingly prevalent in an aging society. Alzheimer’s disease (AD), the most common type of dementia, is best mitigated via early detection and management. Deep learning is an artificial intelligence technique that has been used to diagnose and predict diseases by extracting meaningful features from medical images. The convolutional neural network (CNN) is a representative application of deep learning, serving as a powerful tool for the diagnosis of AD. Recently, vision transformers (ViT) have yielded classification performance exceeding that of CNN in some diagnostic image classifications. Because the brain is a very complex network with interrelated regions, ViT, which captures direct relationships between images, may be more effective for brain image analysis than CNN. Therefore, we propose a method for classifying dementia images by applying 18F-Florbetaben positron emission tomography (PET) images to ViT. Data were evaluated via binary (normal control and abnormal) and ternary (healthy control, mild cognitive impairment, and AD) classification. In a performance comparison with the CNN, VGG19 was selected as the comparison model. Consequently, ViT yielded more effective performance than VGG19 in binary classification. However, in ternary classification, the performance of ViT cannot be considered excellent. These results show that it is hard to argue that the ViT model is better at AD classification than the CNN model.

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