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Crossover phenomena in the critical behavior for long-range models with power-law couplings. (arXiv:1812.10275v2 [math-ph] UPDATED)
来源于:arXiv
This is a short review of the two papers on the $x$-space asymptotics of the
critical two-point function $G_{p_c}(x)$ for the long-range models of
self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$, defined
by the translation-invariant power-law step-distribution/coupling
$D(x)\propto|x|^{-d-\alpha}$ for some $\alpha>0$. Let $S_1(x)$ be the
random-walk Green function generated by $D$. We have shown that
$\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($\alpha>2$) to
Riesz ($\alpha<2$), with log correction at $\alpha=2$;
$\bullet~~G_{p_c}(x)\sim\frac{A}{p_c}S_1(x)$ as $|x|\to\infty$ in dimensions
higher than (or equal to, if $\alpha=2$) the upper critical dimension $d_c$
(with sufficiently large spread-out parameter $L$). The model-dependent $A$ and
$d_c$ exhibit crossover at $\alpha=2$.
The keys to the proof are (i) detailed analysis on the underlying random walk
to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power
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