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.