论文标题
多物种的波动结果Sherrington-Kirkpatrick模型
Fluctuation results for Multi-species Sherrington-Kirkpatrick model in the replica symmetric regime
论文作者
论文摘要
我们研究了一般多种物种的复制对称区域Sherrington-Kirkpatrick（MSK）模型，并回答Ann。〜probab。〜43〜（2015），第6、3494---3513号中提出的一些问题，在\ emph patifection $ emph {$ emph {$ emph {$ emph {$ patife $ cov in cov in cov cov in cov cons of cov in cov in cov in con中} 首先，我们证明了\ indf〜和\ emph {stustry-definite} $δ^2 $ msk模型在高温下的指数重叠浓度。我们还使用重叠浓度证明了自由能的中心极限定理。此外，在零外场情况下，我们使用二次耦合参数证明重叠浓度高达$β_C$，这预计将是关键的反向温度。 \ emph {paster-definite}和emph {nodefinite} $δ^2 $的参数既有，$β_C$在两种不同的情况下具有相同的表达式。 其次，我们开发了一种通过物种腔体的方法来研究重叠波动，并获得了重叠的渐近方差 - 稳态矩阵，作为对基质值线性系统的解决方案。渐近方差还表明了de almeida-从复制对称（RS）侧的无线线条件。我们的物种腔体方法不需要$δ^2 $的正定义。但是，看来\ emph {paster-definite}和\ emph {Intefinite}的情况不同的情况似乎不同。 最后，在\ emph {paster-definite} $δ^2 $的情况下，我们证明在某些自然假设下，MSK模型处于复制对称性破坏阶段。这概括了J.〜Stat。〜Phys。〜174（2019），第2、333--350号的结果，从2个物种到一般物种。
We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under \emph{positive-definite} assumption on the disorder covariance matrix $Δ^2$. First, we prove exponential overlap concentration at high temperature for both \indf~and \emph{positive-definite} $Δ^2$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupling argument to prove overlap concentration up to $β_c$, which is expected to be the critical inverse temperature. The argument holds for both \emph{positive-definite} and emph{indefinite} $Δ^2$, and $β_c$ has the same expression in two different cases. Second, we develop a species-wise cavity approach to study the overlap fluctuation, and the asymptotic variance-covariance matrix of overlap is obtained as the solution to a matrix-valued linear system. The asymptotic variance also suggests the de Almeida--Thouless (AT) line condition from the Replica Symmetry (RS) side. Our species-wise cavity approach does not require the positive-definiteness of $Δ^2$. However, it seems that the AT line conditions in \emph{positive-definite} and \emph{indefinite} cases are different. Finally, in the case of \emph{positive-definite} $Δ^2$, we prove that above the AT line, the MSK model is in Replica Symmetry Breaking phase under some natural assumption. This generalizes the results of J.~Stat.~Phys.~174 (2019), no.~2, 333--350, from 2-species to general species.
