Christophe Garban
We survey in this paper the main contributions of Oded Schramm related to noise sensitivity. We will describe in particular his various works which focused on the "spectral analysis" of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.
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In this note, I will describe some highlights of Oded Schramm's work in circle packings and the Koebe conjecture, as well as on SLE.
Oded Schramm (1961-2008) influenced greatly the development of percolation theory beyond the usual ${\mathbb{Z}}^d$ setting; in particular, the case of nonamenable lattices. Here, we review some of his work in this field.
Proofs of sharp phase transition and noise sensitivity in percolation have been significantly simplified by the use of randomized algorithms, via the OSSS inequality (proved by O'Donnell, Saks, Schramm and Servedio (2005)) and the Schramm-Steif inequality for the Fourier-Walsh coefficients of functions defined on the Boolean hypercube. In this article, we prove intrinsic versions of the OSSS and Schramm-Steif inequalities for functionals of a general Poisson process, and apply these new estimates to deduce sufficient conditions - expressed in terms of randomized stopping sets - yielding sharp …
We show that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm approach introduced by Schramm and Steif and gives quantitative polynomial bounds on the noise sensitivity of crossing events for Bargmann-Fock. A rather counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock Gaussian field in $\mathbb{R}^3$ and consider two horizontal planes $P_1,P_2$ at small distance $\varepsilon$ from each other. Even though $F$ is a.s. analytic, the above noise sensitivity statement implies that the full res…
Consider critical Bernoulli percolation in the plane. We give a new proof of the sharp noise sensitivity theorem shown by Garban, Pete and Schramm. Contrary to the previous approaches, we do not use any spectral tool. We rather study differential inequalities satisfied by a dynamical four-arm event, in the spirit of Kesten's proof of scaling relations. We also obtain new results in dynamical percolation. In particular, we prove that the Hausdorff dimension of the set of times with both primal and dual percolation equals $2/3$ a.s.
During the past 15 years, several extensions of the concept of noise sensitivity first coined in~\cite{schramm2000}, has been studied. One such extension was studied in ??, where the definition of noise sensitivity was extended to noise corresponding to any sequence of irreducible and reversible Markov chains. In this paper we focus on the case where the Markov chain is a random walk on a Schreier graph, and show that the Benjamini-Kalai-Schramm theorem, connecting influences to noise sensitivity, holds in this setting. We then apply this result to give an alternative proof of one of the main …
We state and prove a version for invasion percolation of Schramm and Smirnov's (2001) discrete gluing theorem for critical planar percolation. The result is a first step toward establishing factorization and characterizing noise sensitivity for invasion percolation.
Noise sensitivity in last-passage percolation
本文研究了Kardar-Parisi-Zhang普遍性类中空间生长过程的噪声敏感性，证明了几何最后一层渗透中旅行时间的噪声敏感性。
We introduce the notion of classical fractional query algorithms, which generalize decision trees in the average-case setting, and can potentially perform better than them. We show that the limiting run-time complexity of a natural class of these algorithms obeys the non-linear partial differential equation $\min_{k}\partial^{2}u/\partial x_{k}^{2}=-2$, and that the individual bit revealment satisfies the Schramm-Steif bound for Fourier weight, connecting noise sensitivity with PDEs. We discuss relations with other decision tree results.
During the past 15 years, several extensions of the concepts noise sensitivity and noise stability, first coined in~\cite{schramm2000}, has been studied. The purpose in this paper is to give definitions of this concepts in the setting of continuous time Markov chains, which then unifies many of the previously considered generalizations. In addition, a considerable amount of time is spent on proving the existence of sequences of noise stable and nondegenerate functions with respect to various classes of Markov chains, a problem which interestingly will appear to have close connections to the so…
Enhanced noise sensitivity, 2D directed polymers and Stochastic Heat Flow
本文将噪声敏感性理论推广至一般随机变量，证明二维有向聚合物配分函数在临界状态具有增强噪声敏感性，并推出随机热流独立于无序白噪声。
In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete p…
We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete mod…
Let $\mathcal{H}$ denote a collection of subsets of $\{1,2,\ldots,n\}$, and assign independent random variables uniformly distributed over $[0,1]$ to the $n$ elements. Declare an element $p$-present if its corresponding value is at most $p$. In this paper, we quantify how much the observation of the $r$-present ($r>p$) set of elements affects the probability that the set of $p$-present elements is contained in $\mathcal{H}$. In the context of percolation, we find that this question is closely linked to the near-critical regime. As a consequence, we show that for every $r>1/2$, bond percolation…
The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm [Inst. Hautes Études Sci. Publ. Math. 90 (1999) 5-43], it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability $p_c$. Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise given a specific witness in the original input. Our main context is the Erdős-Rényi random graph, where…
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of squares of influences in $f$ is close to zero then $f$ must be noise sensitive. We show a quantitative version of this result which does not depend on $n$, and prove that it is tight for certain parameters. Our results hold also for a general product measure $μ_p$ on the discrete cube, as long as $\log 1/p \ll \log n$. We note that in [BKS], a quantitative relatio…
We study functions on the infinite-dimensional Hamming cube $\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$, generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for $n\in\mathbb{N}$. The notion of noise sensitivity, first studied in arXiv:math/9811157 , is extended to this setting, and basic Fourier formulas are established. We also prove hypercontractivity estimates for these functions, and give a version of the Kahn-Kalai-Linial theorem giving a bound relating the total influence to the maximal influence. Particular attention is paid to so-called finitary…
Spectra of Poisson functionals and applications in continuum percolation
本文为泊松泛函引入谱点过程概念，并利用其证明临界连续渗流模型（泊松布尔与泊松沃罗诺伊）中交叉事件的尖锐噪声不稳定性和敏感性，核心工具是 Wiener-Itô 展开与拟乘性估计。
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