Survival probability of the branching random walk killed below a linear boundary

We give an alternative proof of a result by N. Gantert, Y. Hu and Z. Shi on the asymptotic behavior of the survival probability of the branching random walk killed below a linear boundary, in the special case of deterministic binary branching and bounded random walk steps. Connections with the Brunet-Derrida theory of stochastic fronts are discussed

An inverse mapping theorem for blow-Nash maps on singular spaces

A semialgebraic map $f:X\to Y$ between two real algebraic sets is called blow-Nash if it can be made Nash (i.e. semialgebraic and real analytic) by composing with finitely many blowings-up with non-singular centers. We prove that if a blow-Nash self-homeomorphism $f:X\rightarrow X$ satisfies a lower bound of the Jacobian determinant condition then $f^{-1}$ is also blow-Nash and satisfies the same condition. The proof relies on motivic integration arguments and on the virtual Poincar\'e polynomial of McCrory-Parusi\'nski and Fichou. In particular, we need to generalize Denef-Loeser change of variables key lemma to maps that are generically one-to-one and not merely birational

Cometary topography and phase darkening

Cometary surfaces can change significantly and rapidly due to the sublimation of their volatile material. Many authors have investigated this evolution; Vincent et al. (2017) have used topographic data from all comets visited by spacecrafts to derive a quantitative model which relates large scale roughness (i.e. topography) with the evolution state of the nucleus for Jupiter Family Comets (JFCs). Meanwhile, ground based observers have published measurements of the phase functions of many JFCs and reported a trend in the phase darkening, with primitive objects showing a stronger darkening than evolved ones). In this paper, we use a numerical implementation of the topographic description by Vincent et al. (2017) to build virtual comets and measure the phase darkening induced by the different levels of macro-roughness. We then compare our model with the values published by Kokotanekova et al. (2018) We find that pure geometric effects like self-shadowing can represent up to 22% of the darkening observed for more primitive objects, and 15% for evolved surfaces. This shows that although physical and chemical properties remain the major contributor to the phase darkening, the additional effect of the topography cannot be neglected

Goussarov-Habiro theory for string links and the Milnor-Johnson correspondence

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $\mu$ invariant. These invariants are seen to be naturally related to invariants of homology cylinders through the so-called Milnor-Johnson correspondence: in particular, an analogue of the Birman-Craggs homomorphism for string links is computed. The relation with Vassiliev theory is studied.Comment: 23 pages. New exposition. One new section (relation with Vassiliev theory). To appear in Topology & its Application

On the affine random walk on the torus

Let $\mu$ be a borelian probability measure on $\mathbf{G}:=\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{T}^d$. Define, for $x\in \mathbb{T}^d$, a random walk starting at $x$ denoting for $n\in \mathbb{N}$, $$\left\{\begin{array}{rcl} X_0 &=&x\\ X_{n+1} &=& a_{n+1} X_n + b_{n+1} \end{array}\right.$$ where $((a_n,b_n))\in \mathbf{G}^\mathbb{N}$ is an iid sequence of law $\mu$. Then, we denote by $\mathbb{P}_x$ the measure on $(\mathbb{T}^d)^\mathbb{N}$ that is the image of $\mu^{\otimes \mathbb{N}}$ by the map $\left((g_n) \mapsto (x,g_1 x, g_2 g_1 x, \dots , g_n \dots g_1 x, \dots)\right)$ and for any $\varphi \in \mathrm{L}^1((\mathbb{T}^d)^\mathbb{N}, \mathbb{P}_x)$, we set $\mathbb{E}_x \varphi((X_n)) = \int \varphi((X_n)) \mathrm{d}\mathbb{P}_x((X_n))$. Bourgain, Furmann, Lindenstrauss and Mozes studied this random walk when $\mu$ is concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes\{0\}$ and this allowed us to study, for any h\"older-continuous function $f$ on the torus, the sequence $(f(X_n))$ when $x$ is not too well approximable by rational points. In this article, we are interested in the case where $\mu$ is not concentrated on $\mathrm{SL}_d(\mathbb{Z}) \ltimes \mathbb{Q}^d/\mathbb{Z}^d$ and we prove that, under assumptions on the group spanned by the support of $\mu$, the Lebesgue's measure $\nu$ on the torus is the only stationary probability measure and that for any h\"older-continuous function $f$ on the torus, $\mathbb{E}_x f(X_n)$ converges exponentially fast to $\int f\mathrm{d}\nu$. Then, we use this to prove the law of large numbers, a non-concentration inequality, the functional central limit theorem and it's almost-sure version for the sequence $(f(X_n))$. In the appendix, we state a non-concentration inequality for products of random matrices without any irreducibility assumption