T-4: Difference between revisions

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1 & q_2 &q_1& q_1 & q_0 \cdots& q_0\\
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_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 &1& q_2 & q_0 \cdots& q_0\\
q_1 & q_1 &q_2& 1 & q_0 \cdots& q_0\\
q_1 & q_1 &q_2& 1 & q_0 \cdots& q_0\\

Revision as of 17:15, 24 December 2023

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 between configurations belonging to the same pure state, 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_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 and . 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: