The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn ² as , where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn as . The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction.     

Improved upper bounds for the critical probability of oriented percolation in two dimensions

We refine the method of our previous paper [2] which gave upper bounds for the critical probability in two-dimensional oriented percolation. We use our refinement to show that © 1994 John Wiley & Sons, Inc.     

Large deviations for the contact process and two dimensional percolation

Summary The following results are proved: 1) For the upper invariant measure of the basic one-dimensional supercritical contact process the density of 1's has the usual large deviation behavior: the probability of a large deviation decays exponentially with the number of sites considered. 2) For supercritical two-dimensional nearest neighbor site (or bond) percolation the densityY of sites inside a square which belong to the infinite cluster has the following large deviation properties. The probability thatY deviates from its expected value by a positive amount decays exponentially with the area of , while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of . These two problems are treated together in this paper because similar techniques (renormalization) are used for both.Partially supported by the National Science Foundation and the U.S. Army Research Office through the Mathematical Sciences Institute at CornellPartially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Brazil) and the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell     

Mean-field critical behaviour for correlation length for percolation in high dimensions

Summary Extending the method of [27], we prove that the corrlation length of independent bond percolation models exhibits mean-field type critical behaviour (i.e. (p(p _c –p)^–1/2 aspp _c ) in two situations: i) for nearest-neighbour independent bond percolation models on ad-dimensional hypercubic lattice ^d , withd sufficiently large, and ii) for a class of spread-out independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents , , , and ₂ for the above two cases.     

A central limit theorem for “critical” first-passage percolation in two dimensions

Summary. Consider (independent) first-passage percolation on the edges of ℤ ² . Denote the passage time of the edge e in ℤ ² by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C ₀ } = 0 for some constant C ₀ >0 and that E[t ^δ (e)]<∞ for some δ>4. Denote by b _0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C ₁ , C ₂ <∞ and γ _n such that C ₁ ( log n) ^1/2 ≦γ _n ≦ C ₂ ( log n) ^1/2 and such that γ _n ⁻¹ [b _0,n −Eb _0,n ] and (√ 2γ _n ) ⁻¹ [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ ² , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C ₀ } = p _c (ℤ ² , site ) := critical probability of site percolation on ℤ ² , and E[t ^δ (u)]<∞ for some δ>4. Received: 6 February 1996 / In revised form: 17 July 1996     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

The exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket

The authors rigorously prove that the exponent for the mean square displacement of self-avoiding random walk on the Sierpinski gasket is

The critical contact process seen from the right edge

Summary Durrett (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to it held starting in any semi-infinite initial state. We prove the same for the critical contact process. We also prove that the process starting with one particle, conditioned to survive until timet, converges to the unique invariant measure ast.Partially supported by the National Science FoundationPartially supported by the National Science Foundation, the National Security Agency, and the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Lagrangian intersections, critical points and Qcategory

For a manifold M, we prove that any function defined on a vector bundle of basis M and quadratic at infinity has at least Qcat(M)+1 critical points. Here Qcat(M) is a homotopically stable version of the LS-category defined by Scheerer, Stanley and Tanré [27]. The key homotopical result is that Qcat(M) can be identified with the relative LS-category of Fadell and Husseini [9] of the pair (M×D ⁿ⁺¹ ,M×S ⁿ ) for n big enough. Combining this result with the work of Laudenbach and Sikorav [19], we obtain that if M is closed, for any hamiltonian diffeomorphism with compact support of T ^* M, #((M)M)Qcat(M)+1, which improves all previously known homotopical estimates of this intersection number. Mathematics Subject Classification (2000):53D12, 55M30, 57R70.     

Rank-based selection strategies for the random walk process

In many decision situations such as hiring a secretary, selling an asset, or seeking a job, the value of each offer, applicant, or choice is assumed to be an independent, identically distributed random variable. In this paper, we consider a special case where the observations are auto-correlated as in the random walk model for stock prices. For a given random walk process of n observations, we explicitly compute the probability that the j-th observation in the sequence is the maximum or minimum among all n observations. Based on the probability distribution of the rank, we derive several distribution-free selection strategies under which the decision maker's expected utility of selecting the best choice is maximized. We show that, unlike in the classical secretary problem, evaluating more choices in the random walk process does not increase the likelihood of successfully selecting the best.     

Critical behavior and the limit distribution for long-range oriented percolation. I

We consider oriented percolation on ${\mathbb{Z}}^d\times{\mathbb{Z}}_+$ whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on ${\mathbb{Z}}^d$ . Suppose that D(x) decays as |x|^?d?α for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension $d_c=2(\alpha\wedge2)$ . We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to $e^{-c|k|^{\alpha\wedge2}}$ for some c > 0.     

Limit theorems for critical first-passage percolation on the triangular lattice

Consider (independent) first-passage percolation on the sites of the triangular lattice $\mathbb{T}$ embedded in $\mathbb{C}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t\left(v\right)$, and assume that $P(t\left(v\right)=0)=P(t\left(v\right)=1)=1∕2$. Denote by $b_{0,n}$ the passage time from 0 to the halfplane $\{v\in \mathbb{T}:\text{Re}\left(v\right)\ge n\}$, and by $T(0,nu)$ the passage time from 0 to the nearest site to $nu$, where $\left|u\right|=1$. We prove that as $n\to \infty$, $b_{0,n}∕logn\to 1∕\left(2\sqrt{3}\pi \right)$ a.s., $E\left[b_{0,n}\right]∕logn\to 1∕\left(2\sqrt{3}\pi \right)$ and Var$\left[b_{0,n}\right]∕logn\to 2∕\left(3\sqrt{3}\pi \right)?1∕\left(2{\pi }^2\right)$; $T(0,nu)∕logn\to 1∕\left(\sqrt{3}\pi \right)$ in probability but not a.s., $E\left[T(0,nu)\right]∕logn\to 1∕\left(\sqrt{3}\pi \right)$ and Var$\left[T(0,nu)\right]∕logn\to 4∕\left(3\sqrt{3}\pi \right)?1∕{\pi }^2$. This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for $b_{0,n}$ and $T(0,nu)$. A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble ${\mathrm{CLE}}_6$, given by Schramm et al. (2009).     

Bounds and majorization relations for the critical points of polynomials

We apply several matrix inequalities to the derivative companion matrices of complex polynomials to establish new bounds and majorization relations for the critical points of these polynomials in terms of their zeros.     

A lower bound for the critical probability of the square lattice site percolation

Summary We prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0.503478.Work supported by the Central Research Found of the Hungarian Academy of Sciences (Grant No. 476/82)