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Register a new userLace Expansion and Mean-Field Behavior for the Random Connection Model
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We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider thr
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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.
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 characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition.
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(alphawedge2)$ for self-avoiding walk and the Ising model, and $d>3(alphawedge2)$ for percolation, where $d$ denotes the dimension and $alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
The main results in this paper are about the full coalescence time $mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $mathbf{E}[mathsf{C}]approx 2mathsf{m}(G)$ and $mathsf{C}/mathsf{m}(G)$ has approximately the same law as in the mean field setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $dgeq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality. Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean $approxmathsf{m}(G)/bigl({matrix{k 2}}bigr)$.
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