T-4

Goal of these problems:

Key concepts:

The free-energy landscape of the SK model

We have seen an example of mean-field model, the spherical p-spin, in which the low-T phase is glassy, described by a 1-RSB ansatz of the overlap matrix. The thermodynamics in the glassy phase is described by three quantities: the typical overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_1^{SP} } between configurations belonging to the same pure state, the typical overlap $q_{0}^{SP}$ between configurations belonging to different pure states, and the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-m^{SP}) } that two configurations extracted at equilibrium belong to the same state. It can be shown that the low-T, frozen phase of the REM is also described by this 1-RSB ansatz with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_0^{SP}=0, q_1^{SP}=1} and $m^{SP}=T/T_{c}$. Replicas are a way to explore the structure of the free-energy landscape.

The Sherrington-Kirkpatrick model introduced in Lecture 1 also has a low-T phase that is glassy. However, the structure of the free-energy landscape is more complicated the mutual overlaps between equilibrium states are organized in a complicated pattern. To understand it, let us consider a 2-RSB ansatz for the overlap matrix:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q=\begin{pmatrix} 1 & q_2 &q_1& q_1 & q_0 \cdots& q_0\\ q_2 & 1 &q_1& q_1 & q_0 \cdots& q_0\\ q_1 & q_1 &1& q_2 & q_0 \cdots& q_0\\ q_1 & q_1 &q_2& 1 & q_0 \cdots& q_0\\ \cdots\\ \cdots\\ \cdots\\ q_0 & q_0 \cdots& q_2&1 & q_1 &q_1\\ q_0 & q_0 \cdots& q_1&q_1 & 1 &q_2\\ q_0 & q_0 \cdots& q_1&q_1 & q_2 &1\\ \end{pmatrix} }

which assumes that replicas are split into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} blocks, and that inside each block they are further split into $m_{2}<m_{1}$ blocks (in the example above,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1=4, m_2=2} ). This is also not the correct structure for the SK model. The correct one obtained iterating this procedure an infinite number of times, as understood by Parisi in his seminal paper of XX. One ends up with a series of overlaps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_K > q_{K-1} > \cdots >q_0} with $K\to \infty$. The underlying picture of the free-energy landscape is as follows: equilibrium states have self-overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_K = q_{EA}} . They are arranged in clusters such that states inside a cluster have overlap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{K−1}} , but such clusters are arranged in other clusters at a higher level, at mutual overlap Failed to parse (syntax error): {\displaystyle q_{K−2}} and so on. In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \to \infty} , the overlap distribution becomes a continuous function.