In this paper, we observe the frog model, an infinite system of interacting random walks, on Z with an asymmetric underlying random walk. Under the assumption of transience with a fixed frog distribution, we construct an explicit formula for the moments of the lower bound of the model’s long-run range, as well as their asymptotic limits as the drift of the underlying random walk vanishes. We al...

We study the first exit time τ from an arbitrary cone with apex at the origin by a non-homogeneous random walk on Z2 with mean drift that is asymptotically zero. Assuming bounded jumps and a form of weak isotropy, we give conditions for τ to be almost surely finite, and for the existence and non-existence of moments E[τp], p > 0. Specifically, if the mean drift at x ∈ Z2 is of magnitude O(‖x‖−1...

One controversy about the existence of so called evolutionary forces such as natural selection and random genetic drift concerns the sense in which such “forces” can be said to interact. In this paper I explain how natural selection and random drift can interact. In particular, I show how population-level probabilities can be derived from individual-level probabilities, and explain the sense in...

Analyzing functional magnetic resonance imaging (fMRI) pattern similarity is becoming increasingly popular because it allows one to relate distributed patterns of voxel activity to continuous perceptual and cognitive states of the human brain. Here we show that fMRI pattern similarity estimates are severely affected by temporal pattern drifts in fMRI data – even after voxelwise detrending. For ...