Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Crossover phenomena in the critical behavior for long-range models with
  power-law couplings   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
functions (with log corrections, if $\alpha=2$) to optimally control the
lace-expansion coefficients $\pi_p^{(n)}$, and (iii) probabilistic
interpretation (valid only when $\alpha\le2$) of the convolution of $D$ and a
function $\varPi_p$ of the alternating series
$\sum_{n=0}^\infty(-1)^n\pi_p^{(n)}$. We outline the proof, emphasizing the
above key elements for percolation in particular.

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The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
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The categories of this paper are: eess
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The categories of this paper are: stat