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Renewal structure of the Tree Builder Random Walk
Comments: In this version, made several improvements in terms of readability. We also have a new title which better reflects the contributions of the paper. 31 pages, 6 figures. Comments are always welcomeCategories: math.PRIn this paper, we study a class of random walks that build their own tree. At each step, the walker attaches a random number of leaves to its current position. The model can be seen as a subclass of the Random Walk in Changing Environments (RWCE) introduced by G. Amir, I. Benjamini, O. Gurel-Gurevich and G. Kozma. We develop a renewal framework for the process analogous to that established by A-S. Sznitman and M. Zerner in the context of RWRE. This provides a more robust foundation for analyzing the model. As a result of our renewal framework, we estabilish several limit theorems for the walker's distance, which include the Strong Law of Large Numbers (SLLN), the Law of the Iterated Logarithm (LIL), and the Invariance Principle, under an i.i.d. hypothesis for the walker's leaf-adding mechanism. Further, we show that the limit speed defined by the SLLN is a continuous function over the space of probability distributions on $\mathbb{N}$.
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arXiv:2109.05582 (Published 2021-09-12)
Record statistics of integrated random walks and the random acceleration process
Comments: 32 pages, 7 figuresCategories: cond-mat.stat-mech, math.PRWe address the theory of records for integrated random walks with finite variance. The long-time continuum limit of these walks is a non-Markov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated random walks. This result is universal, i.e., independent of the details of the parent distribution of the step lengths.
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arXiv:1906.07913 (Published 2019-06-19)
Differentiability of the speed of biased random walks on Galton-Watson trees
Comments: 29 pages, 1 figureCategories: math.PRWe prove that the speed of a $\lambda$-biased random walk on a supercritical Galton-Watson tree is differentiable for $\lambda$ such that the walk is ballistic and obeys a central limit theorem, and give an expression of the derivative using a certain $2$-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
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arXiv:1811.04849 (Published 2018-11-12)
Regularity results of the speed of biased random walks on Galton-Watson trees
Categories: math.PRWe prove that the speed of $\lambda$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $\lambda\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
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arXiv:1506.02895 (Published 2015-06-09)
Renewal structure and local time for diffusions in random environment
Comments: 39 pagesCategories: math.PRWe study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with $ 0\textless{}\kappa\textless{}1$, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time $t\textgreater{}0$. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable L{\'e}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.
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Renewal theory for random walks on surface groups
Comments: revised version, to appear in ETDSDOI: 10.1017/etds.2016.15Categories: math.PRTags: journal articleWe construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enters a particular type of a cone and never leaves it again. As a consequence, the trajectory of the random walk can be expressed as an "aligned union" of i.i.d. trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.