Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk $\mathbf{S}_n$ on $\mathbb{Z}^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, $i.e.$ a sharp asymptotic of the Green function $G(\mathbf{0},\mathbf{x})$ as $\|\mathbf{x}\|\to +\infty$, along the "favorite direction or scaling": (i) if $\sum_{i=1}^d \alpha_i^{-1} < 2$ (reminiscent of Garsia-Lamperti's condition when $d=1$ [Comm. Math. Helv. $\mathbf{37}$, 1962]); (ii) if a certain $local$ condition holds (reminiscent of Doney's condition [Probab. Theory Relat. Fields $\mathbf{107}$, 1997] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0},\mathbf{x})$, sharpening estimates when $\mathbf{x}$ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha_i\equiv \alpha$, in the favorite scaling, and has even left aside the case $\alpha\in[1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.Comment: 46 pages, comments are welcom

THE SETTING OF MILTON\u27S EIGHTEENTH SONNET

...But this sonnet immortalizes an aspect, perhaps only an insignificant aspect of that age; yet its greatness lies less in its subject, which is hardly a universal one, which is, in fact, little more than the angered outburst of a passing moment, than in the splendour and rolling grandeur of its words. Who can show a finer example of emotion recollected in tranquility

Pinning model in random correlated environment: appearance of an infinite disorder regime

We study the influence of a correlated disorder on the localization phase transition in the pinning model. When correlations are strong enough, a strong disorder regime arises: large and frequent attractive regions appear in the environment. We present here a pinning model in random binary ({-1,1}-valued) environment. Defining strong disorder via the requirement that the probability of the occurrence of a large attractive region is sub-exponential in its size, we prove that it coincides with the fact that the critical point is equal to its minimal possible value. We also stress that in the strong disorder regime, the phase transition is smoother than in the homogeneous case, whatever the critical exponent of the homogeneous model is: disorder is therefore always relevant. We illustrate these results with the example of an environment based on the sign of a Gaussian correlated sequence, in which we show that the phase transition is of infinite order in presence of strong disorder. Our results contrast with results known in the literature, in particular in the case of an IID disorder, where the question of the influence of disorder on the critical properties is answered via the so-called Harris criterion, and where a conventional relevance/irrelevance picture holds.Comment: 27 pages, some corrections made in v

Comments on the Influence of Disorder for Pinning Model in Correlated Gaussian Environment

We study the random pinning model, in the case of a Gaussian environment presenting power-law decaying correlations, of exponent decay a>0. We comment on the annealed (i.e. averaged over disorder) model, which is far from being trivial, and we discuss the influence of disorder on the critical properties of the system. We show that the annealed critical exponent \nu^{ann} is the same as the homogeneous one \nu^{pur}, provided that correlations are decaying fast enough (a>2). If correlations are summable (a>1), we also show that the disordered phase transition is at least of order 2, showing disorder relevance if \nu^{pur}<2. If correlations are not summable (a<1), we show that the phase transition disappears.Comment: 23 pages, 1 figure Modifications in v2 (outside minor typos): Assumption 1 on correlations has been simplified for more clarity; Theorem 4 has been improved to a more general underlying renewal distribution; Remark 2.1 added, on the assumption on the correlations in the summable cas

Singular measure traveling waves in an epidemiological model with continuous phenotypes

We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+\mu\left(\int_\Omega M(y,z)u(t,x,z)dz-u\right) + u\left(a(y)-\int_\Omega K(y,z) u(t,x,z)dz\right) , \end{equation*} where $u=u(t,x,y)$ stands for the density of a theoretical population with a spatial ($x\in\mathbb R$) and phenotypic ($y\in\Omega\subset \mathbb R^n$) structure, $M(y,z)$ is a mutation kernel acting on the phenotypic space, $a(y)$ is a fitness function and $K(y,z)$ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation, and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $c^*:=2\sqrt{-\lambda_1}$, where $\lambda_1$ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation