连续对称Dyson分层模型中的类Thouless-like效应
A Thouless-Like Effect in the Dyson Hierarchical Model with Continuous Symmetry
摘要：
我们研究了Dyson的经典的具有长程相互作用势$U(i,j)= -l(d(i,j)) d^{-2}(i,j)$的$r$组分铁磁分级模型，其中$d(i,j)$表示分级距离.本文证明了Dyson的一个猜想，即级数$l_1+l_2+...$的收敛性（其中$l_n=l(2^n)$）是所考虑的模型中存在相变的充分必要条件，并且在临界点处自发磁化消失，即不存在无磁效应.但我们发现，当体积趋于无穷大时，在临界温度$T_c$下的归一化平均自旋分布趋于$\Bbb R^r$中单位球上的均匀分布，这一现象类似于无量纲效应.
英文摘要：
We study Dyson's classical $r$-component ferromagnetic hierarchical model with a long range interaction potential $U(i,j)= -l(d(i,j)) d^{-2}(i,j)$, where $d(i,j)$ denotes the hierarchical distance. We prove a conjecture of Dyson, which states that the convergence of the series $l_1+l_2+...$, where $l_n=l(2^n)$, is a necessary and sufficient condition of the existence of phase transition in the model under consideration, and the spontaneous magnetization vanishes at the critical point, i.e. there is no Thouless' effect. We find however that the distribution of the normalized average spin at the critical temperature $T_c$ tends to the uniform distribution on the unit sphere in $\Bbb R^r$ as the volume tends to infinity, a phenomenon which resembles the Thouless effect.
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