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تسجيل مستخدم جديدA survey on the lace expansion for the nearest-neighbor models on the BCC lattice
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The aim of this survey is to explain, in a self-contained and relatively beginner-friendly manner, the lace expansion for the nearest-neighbor models of self-avoiding walk and percolation that converges in all dimensions above 6 and 9, respectively. To achieve this, we consider a $d$-dimensional version of the body-centered cubic (BCC) lattice, on which it is extremely easy to enumerate various random-walk quantities. Also, we choose a particular set of bootstrapping functions, by which a notoriously complicated part of the lace-expansion analysis becomes rather transparent.
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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.
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 sp
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In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the $d$-dimensional body-centered cubic (BCC) lattice $mathbb{L}^d$ and the set of non-negative integers $mathbb{Z}_+$. Thanks
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We calculate the local Green function for a quantum-mechanical particle with hopping between nearest and next-nearest neighbors on the Bethe lattice, where the on-site energies may alternate on sublattices. For infinite connectivity the renormalized
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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.
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