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Register a new userDiagrammatic bounds on the lace-expansion coefficients for oriented percolation
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Akira Sakai
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We provide a complete proof of the diagrammatic bounds on the lace-expansion coefficients for oriented percolation, which are used in [arXiv:math/0703455] to investigate critical behavior for long-range oriented percolation above 2min{alpha,2} spatial dimensions.
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We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the $n$-component $g|varphi|^4$ model on $mathbb{Z}^{d}$ when $n=1,2$, and prove that the critical Greens function $G_{ u_{c}}(x)$ is asymptotically a multiple of $|x|^{2-d}$ when $dgeq 5$ at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.
The lace expansion for the Ising two-point function was successfully derived in Sakai (Commun. Math. Phys., 272 (2007): 283--344). It is an identity that involves an alternating series of the lace-expansion coefficients. In the same paper, we claimed that the expansion coefficients obey certain diagrammatic bounds which imply faster $x$-space decay (as the two-point function cubed) above the critical dimension $d_c$ ($=4$ for finite-variance models), if the spin-spin coupling is ferromagnetic, translation-invariant, summable and symmetric with respect to the underlying lattice symmetries. However, we recently found a flaw in the proof of Lemma 4.2 in Sakai (2007), a key lemma to the aforementioned diagrammatic bounds. In this paper, we no longer use the problematic Lemma 4.2 of Sakai (2007), and prove new diagrammatic bounds on the expansion coefficients that are slightly more complicated than those in Proposition 4.1 of Sakai (2007) but nonetheless obey the same fast decay above the critical dimension $d_c$. Consequently, the lace-expansion results for the Ising and $varphi^4$ models so far are all saved. The proof is based on the random-current representation and its source-switching technique of Griffiths, Hurst and Sherman, combined with a double expansion: a lace expansion for the lace-expansion coefficients.
Consider nearest-neighbor oriented percolation in $d+1$ space-time dimensions. Let $rho,eta, u$ be the critical exponents for the survival probability up to time $t$, the expected number of vertices at time $t$ connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $d ugeeta+2rho$, which holds for all $dge1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d=1$, i.e., $ u=eta+2rho$, provided existence of at least two among $rho,eta, u$. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin, Tassion and Teixeira (2017).
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index alpha>0 converges to e^{-C|k|^{alphawedge2}} for some Cin(0,infty) above the upper-critical dimension 2(alphawedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at alpha=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.
Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.
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