It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $α/2n$, exhibits a sharp threshold at $α=1$. We study a more general 2-SAT model in which each clause occurs independently but with probability $α_i/2n$ where $i \in \{0,1,2\}$ is the number of positive literals in that clause. We generalize branching process arguments by Verhoeven(99) to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix.
We analyse the splitting algorithm performance in the estimation of rare event probabilities and this in a discrete multidimensional framework. For this we assume that each threshold is partitioned into disjoint subsets and the probability for a particle to reach the next threshold will depend on the starting subset. A straightforward estimator of the rare event probability is given by the proportion of simulated particles for which the rare event occurs. The variance of this estimator we get is the sum of two parts: one part resuming the variability due to each threshold and a second part res…
Interesting posts are continually forwarded by the users of the online social network (OSN). Such propagation leads to re-forwarding of the post to some of the previous recipients, which increases as the post reaches a large number of users. Consequently, the effective forwards (after deleting the re-forwards) reduce, eventually leading to the saturation of the total number of copies. We model this process as a new variant of the branching process, the `saturated total-population-dependent branching process', and analyse it using the stochastic approximation technique. Notably, we obtain deter…
We define a graph process $\mathcal{G}(p,q)$ based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube $Q_d$ and the lattice $\mathbb{Z}^d$ for large $d$. Individuals have Poisson offspring distribution with mean $1+p$ and certain deletions and mergers occur with probability $q$; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively. We prove survival a…
In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $ξ= \left(ξ(x)\right)_{x\in\mathbb{R}}$ satisfying certain conditions. We show that the maximum position $M_t$ of particles alive at time $t$ satisfies a quenched strong law of large numbers and an annealed invariance principle.
Time-inhomogeneous N-particle Branching Brownian Motion and the continuous random energy model
Università degli Studi di Padova（帕多瓦大学）；Istituto Nazionale di Alta Matematica（高等数学研究所）；Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学）；Institut Universitaire de France（法兰西学院）
We prove the existence and pathwise uniqueness of the solution to a stochastic integral equation driven by Poisson random measures based on Kuznetsov measures for a continuous-state branching process. That gives a direct construction of the sample path of a continuous-state branching process with dependent immigration. The immigration rates depend on the population size via some functions satisfying a Yamada--Watanabe type condition. We only assume the existence of the first moment of the process. The existence of excursion law for the continuous-state branching process is not required. By spe…
Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.
We study the speed of extinction of continuous state branching processes in a Lévy environment, where the associated Lévy process oscillates. Assuming that the Lévy process satisfies the Spitzer's condition and the existence of some exponential moments, we extend recent results where the associated branching mechanism was stable. Our study relies on the path analysis of the process together with its environment, when this latter is conditioned to have a non negative running infimum. This approach is inspired from the discrete setting with i.i.d. environment studied in (Afanasyev et al. 2005).
A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet $(σ^2,a,Λ)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(σ^2,a,Λ)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of…
Maximal Displacement of Critical Branching Symmetric Stable Processes
我们考虑一个临界连续时间分支过程（一个Yule过程），其中的个人独立地执行对称$α-$稳定的随机运动的实数线开始在他们的出生点。因为分支过程是关键的，它最终会消亡，所以存在一个定义明确的最大位置$M$，该过程的单个粒子曾经访问过。证明了$M$的分布满足渐近关系$P\{M\geq x \}\sim (2/α)^{1/2}x^{-α/2}$ as $x \rightarrow \infty$ .
Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension of the failed boundary
We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main resu…
A multitype continuous-state branching process (MCSBP) ${\rm Z}=({\rm Z}_{t})_{t\geq 0}$, is a Markov process with values in $[0,\infty)^{d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{R}^d$-valued spectrally positive Lévy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing…
We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow ρ,\ n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $ρ\in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $ρ=0$ or $1$. We find the a…
One of the most studied models of SAT is random SAT. In this model, instances are composed from clauses chosen uniformly randomly and independently of each other. This model may be unsatisfactory in that it fails to describe various features of SAT instances, arising in real-world applications. Various modifications have been suggested to define models of industrial SAT. Here, we focus mainly on the aspect of community structure. Namely, here the set of variables consists of a number of disjoint communities, and clauses tend to consist of variables from the same community. Thus, we suggest a m…
Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems [Mezard, Parisi, Zecchina: Science 2002]. The cavity method predicts that the satisfiability threshold in the random $k$-SAT problem is $2^k\ln2-\frac12(1+\ln 2)+ε_k$, with $\lim_{k\rightarrow\infty}ε_k=0$ [Mertens, Mezard, Zecchina: Random Structures and Algorithms 2006]. This paper contains a proof of that conjecture.
Viral Evolution and Adaptation as a Multivariate Branching Process
在本工作中，我们分析了RNA病毒群体的适应和进化问题，通过将基本随机模型定义为与Demetrius、Schuster和Sigmund在他们的多核苷酸进化研究中提出的分支过程密切相关的多变量分支过程（“Polynucleotide evolution and branching processes”，Bull.Math.Biol.46（1985）239-262）。我们表明，在没有有利的力量的模型是完全可解的。因此，有可能证明与这些系统的已知典型性质直接相关的几个关键结果，如（i）在分支过程理论的背景下，Bull，Sanjuán和Wilke提出的致死突变标准的证明（“Theory of lethal mutagenesis for viruses”，J. Virology 18（2007）2930-2939）;（ii）关于弛豫时间概念的新建议，并对其进行定量评价;在四个不同的制度：瞬态，“稳定”的平衡，灭绝阈值和致死突变的预期值的演变的定量描述。此外，可以预见对病毒种群演变动态的新见解。
