Akimichi Takemura, Vladimir Vovk, Glenn Shafer
We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.
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Under the unifying umbrella of a general result of Penrose & Yukich [Ann. Appl. Probab., (2003) 13, 277--303] we give laws of large numbers (in the $L^p$ sense) for the total power-weighted length of several nearest-neighbour type graphs on random point sets in $\R^d$, $d\in\N$. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour grap…
We prove that multidimensional diffusions in random environment have a limiting velocity which takes at most two different values. Further, in the two-dimensional case we show that for any direction, the probability to escape to infinity in this direction equals either zero or one. Combined with our results on the limiting velocity, this implies a strong law of large numbers in two dimensions.
Continuing from arXiv:2102.01917v2, in this paper, we discuss general criteria and forms of liminf laws of iterated logarithm (LIL) for continuous-time Markov processes. Under some minimal assumptions, which are weaker than those in arXiv:2102.01917v2, we establish liminf LIL at zero (at infinity, respectively) in general metric measure spaces. In particular, our assumptions for liminf law of LIL at zero and the form of liminf LIL are truly local so that we can cover highly space-inhomogenous cases. Our results cover all examples in arXiv:2102.01917v2 including random conductance models with l…
Khinchin's and Chung's Laws of the Iterated Logarithm at Time Zero for the Linear Stochastic Fractional Diffusion Equation
该研究针对带零初值的线性随机分数阶扩散方程，建立了时间零点的辛钦型重对数律，在特定附加条件下证明了钟氏型重对数律，将随机热方程的初始时间重对数律推广至一类时间分数阶随机扩散方程。
A set of binary random variables indexed by a lattice torus is considered. Under a mixing hypothesis, the probability of any proposition belonging to the first order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to $-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given.
The random key graph is a random graph naturally associated with the random key predistribution scheme of Eschenauer and Gligor for wireless sensor networks. For this class of random graphs we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here complement and strengthen recent work on this conjecture by Blackburn and Gerke. In particular, the results are given under conditions which are more realistic for applications to wireless sensor networks.
We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.
One of the most well-known classical results for site percolation on the square lattice is the equation p_c + p_c^* = 1. In words, this equation means that for all values different from p_c of the parameter p the following holds: Either a.s. there is an infinite open cluster or a.s. there is an infinite closed `star' cluster. This result is closely related to the percolation transition being sharp: Below p_c the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distrubtion with an exponential tail. The analog of this result has been proved by Higuchi in 1993 for t…
One of the most well-known classical results for site percolation on the square lattice is the equation $p_c+p_c^*=1$. In words, this equation means that for all values $\neq p_c$ of the parameter $p$, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed "star" cluster. This result is closely related to the percolation transition being sharp: below $p_c$, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-…
For any fixed positive integer $k$, let $α_{k}$ denote the smallest $α\in (0,1)$ such that the random graph sequence $\left\{G\left(n, n^{-α}\right)\right\}$ does not satisfy the zero-one law for the set $\mathcal{E}_{k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on $α_{k}$, showing that as $k \rightarrow \infty$, we have $α_{k} = \left(k - 2 - t(k)\right)^{-1}$ for some function $t(k) = Θ(k^{-2})$. We also establish the precise value of $α_{k}$ when $k = 4$.
Zero-one laws for events with positional symmetries
该研究用信息论方法证明了充分对称的随机事件满足零一律，该结论可放宽独立同分布条件，既涵盖经典结果也适用于无穷随机图等新场景。
In this paper, we discuss the laws of the iterated logarithm (LIL) for occupation times of Markov processes $Y$ in general metric measure space both near zero and near infinity under some minimal assumptions. We first establish LILs of (truncated) occupation times on balls $B(x,r)$ of radii $r$ up to an function $Φ(r)$, which is an iterated logarithm of mean exit time of $Y$, by showing that the function $Φ$ is optimal. Our first result on LILs of occupation times covers both near zero and near infinity regardless of transience and recurrence of the process. Our assumptions are truly local in …
Infinite products and zero-one laws in categorical probability
本文在马尔可夫范畴框架下发展无穷张量积与柯尔莫哥洛夫扩张定理，并据此证明柯尔莫哥洛夫及休伊特-萨维奇零一律的一般范畴版本，从而将经典测度论结果推广到更广泛的概率语境。
We consider transient random walks in random environment on $\z$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincar{é} invariance. In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions (corresponding respectively to breakings of reflection-invariance, translation-invariance, and Lorentz invariance). Along the way we provide a proof from first principles of the zero-one law on whic…
We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the n sensors in the network is classified into r classes according to some probability distribution μ = {μ_1 , . . . , μ_r }. Before deployment, a class-i sensor is assigned K_i cryptographic keys that are selected uniformly at random from a common pool of P keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scal…
Borel-Cantelli, zero-one laws and inhomogeneous Duffin-Schaeffer
本文建立了适用于任意概率空间且保证 limsup 集概率为 1 的 Borel-Cantelli 引理新版本，并据此解决了非齐次 Duffin-Schaeffer 猜想等问题，提出了动力系统中的新刻画与引理。
An approximate zero-one law via the Dialectica interpretation
本文受哥德尔Dialectica解释启发，将句法零一律有限化为近似零一律，使其完全以事件的有限并交表述，并赋予计算解释以支持定理的定量分析。
Rotated semicircle laws for permanental roots of Gaussian random matrices
本研究针对标准GOE与GUE矩阵，证明其积和式特征多项式的归一化零点计数测度几乎必然收敛于旋转π/2后的Wigner半圆律，验证了Fyodorov提出的相关猜想。
Limit laws for component-pruned sparse random graphs and percolated tori
研究组件修剪稀疏随机图和渗流环面的极限定律，通过结合均匀组件计数等方法证明特定条件下的MSO₂零一律，还探讨了相关项不可省略的情况及建立了离散环面键渗流的一阶极限定律等。
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