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In this article, we consider several models of random walks in one or severaldimensions, additionally allowing, at any unit of time, a reset (or"catastrophe") of the walk with probability $q$. We establish the distributionof the final altitude. We prove algebraicity of the generating functions ofwalks of bounded height $h$ (showing in passing the equivalence betweenLagrange interpolation and the kernel method). To get these generatingfunctions, our approach offers an algorithm of cost $O(1)$, instead of cost$O(h^3)$ if a Markov chain approach would be used. The simplest nontrivialmodel corresponds to famous dynamics in population genetics: the Moran model.  We prove that the height of these Moran walks asymptotically follows adiscrete Gumbel distribution. For $q=1/2$, this generalizes a model of carrypropagation over binary numbers considered e.g. by von Neumann and Knuth. Forgeneric $q$, using a Mellin transform approach, we show that the asymptoticheight exhibits fluctuations for which we get an explicit description (and, inpassing, new bounds for the digamma function). We end by showing how to solvemultidimensional generalizations of these walks (where any subset of particlesis attributed a different probability of dying) and we give an application tothe soliton wave model.

Height of walks with resets, the Moran model, and the discrete Gumbel distribution