Yacine Aoun
The purpose of this modest note is to provide a short proof of the sharpness of the phase transition for the Random-cluster model with $q=2$ by extending the approach developed by Duminil-Copin and Tassion for $q=1$. This in particular implies the exponential decay of the two point-correlation function in the subcritical Ising model.
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We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. - for $p>p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $θ(p)\ge\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework o…
We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $β<β_c$, and the mean-field lower bound $\mathbb{P}_β[0\longleftrightarrow\infty]\ge (β-β_c)/β$ for $β>β_c$. For finite-range models, we also prove that for any $β<β_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we pro…
Sharpness of Bernoulli percolation via couplings
本文利用受 Russo 启发的随机耦合控制，在一般传递无限图上建立 Bernoulli 渗流连接概率估计，无需微分公式即证明相变锐性，并导出次临界指数衰减与超临界平均场下界。
In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive g…
Supercritical sharpness for Ising double random currents
该研究证明了d≥3时伊辛双随机电流渗流模型的超临界尖锐性，简化了相关证明并得到伊辛模型截断关联指数衰减等应用结果。
We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the …
We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…
Double cluster swapping for spin models: Pfaffian relations and sharpness
该研究提出自旋模型的双簇交换几何表示，证明Pfaffian关系蕴含平面性，并推广经典论证证明一类自旋模型相变的尖锐性，兼具理论意义与方法通用性。
In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability $p$, independently for every vertex, every neighbouring edge is opened. We study the size of open clusters in this model. Hirsch et al. have shown that for small $p$ all open clusters are finite almost surely, while for large $p$, depending on the underlying graph, there exists an infinite open cluster almost surely. We show that the corresponding phase transition is sharp, i.e., in the subcritical regime, all open clusters ar…
Sharpness of the phase transition for constrained-degree percolation
本文研究受限度渗透的相变尖锐性，通过随机算法方法证明了一臂概率在亚临界相中指数衰减，并解决了相关问题。
Sharpness of the percolation phase transition for weighted random connection models
本文研究加权随机连接模型中渗透相变的锐度，通过证明亚临界和超临界区域的簇大小分布特性，揭示了临界行为的数学结构。
Percolation of strongly correlated Gaussian fields II. Sharpness of the phase transition
本文证明了强相关高斯场逾渗模型相变的尖锐性，涵盖离散高斯自由场、膜模型等，不依赖正相关性，并利用弱混合性质为平面正相关场建立更优的逾渗密度界及盒子穿越性质。
We present a short proof of the Kac-Ward formula for the partition function of the Ising model on planar graphs.
Sharp phase transition for percolation with short-range dependencies
研究\(\mathbb{Z}^d\)上最近邻渗流模型相变的尖锐性，用OSSS不等式及改进方法证明，还对伯努利键渗流和位点渗流相结合的特殊情况做更广泛研究。
We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence and satisfies a local boundedness condition, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction.
We provide a short proof of the intriguing characterisation of the convex order given by Wiesel and Zhang.
We expose here a short proof of Cramer's theorem in R based on convex duality.
A short proof of Feige's conjecture for identically distributed random variables
针对独立同分布随机变量场景，本文给出Feige猜想的简短证明，为该猜想的验证提供了简洁方法。
A short proof of diffusivity for the directed polymers in the weak disorder phase
本文针对弱无序相中随机环境下的有向聚合物，给出了扩散性的一个新的简短初等证明。
By using the properties of the uniformly distributed sequences of real numbers on $(0,1)$, a short proof of a certain version of Kolmogorov strong law of large numbers is presented which essentially differs from Kolmogorov's original proof.
There is a serious mistake in the proof.
A short proof of Isserlis' theorem
本文通过将Isserlis定理作为各向同性张量不变张量定理的推论，给出了该定理的一个简短证明。
We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Prékopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.
An elementary proof of the dual representation of Expected Shortfall
本文在一般概率空间上利用分位数函数的基本性质，给出了Expected Shortfall对偶表示的初等证明，并由此推导出其次可加性的新证明。
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