T-3 - Revision history

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:33:02Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:33, 13 February 2026
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	\mathbb{E}[P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),		\mathbb{E}[P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),
	</math>		</math>
	The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \mathbb{E}[P_\beta (q)]</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:		The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \mathbb{E}[P_\beta (q)]</math>. Moreover, the Edwards-Anderson order parameter introduced by Alberto in Lecture 1 can also be read out from the formalism:
	<math display="block">		<math display="block">
	q_{EA}= \max \left\{ q_{a \neq b}^{*} \right\}.		q_{EA}= \max \left\{ q_{a \neq b}^{*} \right\}.

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:32:45Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:32, 13 February 2026
Line 119:		Line 119:
	\mathbb{E}[P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),		\mathbb{E}[P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),
	</math>		</math>
	The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \~~overline~~{P_\beta (q)}</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:		The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \mathbb{E}[P_\beta (q)]</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:
	<math display="block">		<math display="block">
	q_{EA}= \max \left\{ q_{a \neq b}^{*} \right\}.		q_{EA}= \max \left\{ q_{a \neq b}^{*} \right\}.

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:32:23Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:32, 13 February 2026
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	This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely, one can show that		This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely, one can show that
	<math display="block">		<math display="block">
	\mathbb{E][P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),		\mathbb{E}[P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),
	</math>		</math>
	The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \overline{P_\beta (q)}</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:		The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \overline{P_\beta (q)}</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:32:13Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:32, 13 February 2026
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	P_{N, \beta}(q)= \int_{\mathcal{S}_N} d \vec{\sigma} \, \int_{\mathcal{S}_N} d \vec{\tau} \frac{e^{-\beta E(\vec{\sigma})}}{Z}\frac{e^{-\beta E(\vec{\tau})}}{Z}\delta \left(q- q(\vec{\sigma}, \vec{\tau}) \right)		P_{N, \beta}(q)= \int_{\mathcal{S}_N} d \vec{\sigma} \, \int_{\mathcal{S}_N} d \vec{\tau} \frac{e^{-\beta E(\vec{\sigma})}}{Z}\frac{e^{-\beta E(\vec{\tau})}}{Z}\delta \left(q- q(\vec{\sigma}, \vec{\tau}) \right)
	</math>		</math>
	This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely,		This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely, one can show that
	<math display="block">		<math display="block">
	\mathbb{E][P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),		\mathbb{E][P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:31:52Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:31, 13 February 2026
Line 117:		Line 117:
	This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely,		This is the distribution of overlaps between configurations extracted at equilibrium, in a fixed given realisation of disorder. On the other hand, the problems above show that replicas play exactly this game: they are different configurations of the system, extracted with a Boltzmann weight at a given realisation of the random energy landscape. They serve as probes for capturing correlations between equilibrium configurations. The saddle point solution <math> q_{ab}^* </math> capture precisely the information on the average distribution of overlaps in the system. More precisely,
	<math display="block">		<math display="block">
	\~~overline~~{P_\beta(q)}=\lim_{N \to \infty} \~~overline~~{P_{N, \beta}(q)}=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),		\mathbb{E][P_\beta(q)]=\lim_{N \to \infty} \mathbb{E}[P_{N, \beta}(q)]=\lim_{n \to 0} \; \frac{2}{n(n-1)}\; \sum_{a>b}\delta \left(q- q_{ab}^{*}\right),
	</math>		</math>
	The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \overline{P_\beta (q)}</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:		The solution of the saddle point equations for the overlap matrix <math> Q</math> thus contain the information on whether spin glass order emerges, which corresponds to having a non-trivial distribution <math> \overline{P_\beta (q)}</math>. Moreover, the Edwards-Anderson order parameter introduced in Lecture 1 can also be read out from the formalism:

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:31:12Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:31, 13 February 2026
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	== The physics within the replica formalism: a first hint==		== The physics within the replica formalism: a first hint==
	In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,</math>-spin model takes the form:		In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p</math>-spin model takes the form:

	<math display="block">		<math display="block">

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:31:05Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:31, 13 February 2026
Line 110:		Line 110:

	== The physics within the replica formalism: a first hint==		== The physics within the replica formalism: a first hint==
	In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,<\math>-spin model takes the form:		In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,</math>-spin model takes the form:

	<math display="block">		<math display="block">

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:30:59Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:30, 13 February 2026
Line 110:		Line 110:

	== The physics within the replica formalism: a first hint==		== The physics within the replica formalism: a first hint==
	In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,\math>-spin model takes the form:		In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,<\math>-spin model takes the form:

	<math display="block">		<math display="block">

Ros: /* The physics within the replica formalism: a first hint */

2026-02-13T12:30:50Z

The physics within the replica formalism: a first hint

← Older revision		Revision as of 14:30, 13 February 2026
Line 110:		Line 110:

	== The physics within the replica formalism: a first hint==		== The physics within the replica formalism: a first hint==
	In Problem 2, we have introduced the overlap distribution:		In Problem 2, we have introduced the overlap distribution, that for the spherical <math> p,\math>-spin model takes the form:

	<math display="block">		<math display="block">

Ros: /* Problems */

2026-02-07T17:05:59Z

Problems

← Older revision		Revision as of 19:05, 7 February 2026
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	== Problems==		== Problems==

	<!!--=== Problem 3.1: correlations, p-spin vs REM ===		<!-- === Problem 3.1: correlations, p-spin vs REM ===
	<br>		<br>
	At variance with the REM, in the spherical <math>p</math>-spin the energies at different configurations are correlated. Show that		At variance with the REM, in the spherical <math>p</math>-spin the energies at different configurations are correlated. Show that
Line 45:		Line 45:
	</math>		</math>
	is the overlap between the two configurations. Why can we say that for <math>p \to \infty </math> this model converges with the REM?		is the overlap between the two configurations. Why can we say that for <math>p \to \infty </math> this model converges with the REM?
	<br>		<br>-->
	=== Problem 3.2: the annealed free energy ===
			<!--=== Problem 3.2: the annealed free energy ===
	As a preliminary exercise, we compute the annealed free energy of the spherical <math>p</math>-spin model.		As a preliminary exercise, we compute the annealed free energy of the spherical <math>p</math>-spin model.