T-2 - Revision history

Ros: /* Check out: key concepts and exercises */

2026-02-15T18:01:45Z

Check out: key concepts and exercises

← Older revision		Revision as of 20:01, 15 February 2026
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	Freezing transition, typical vs average, domination by rare events, entropy crisis, condensation and extreme events, overlap distribution.		Freezing transition, typical vs average, domination by rare events, entropy crisis, condensation and extreme events, overlap distribution.

	After this lecture, you have all the tools to solve <code>Exercise 5 </code> and <code>Exercise 6 </code>		After this lecture, you have all the tools to solve <code>Exercise 5 </code> and <code>Exercise 6 </code>.

			We go back to the overlap distribution for the REM in <code>Exercise 13 </code>.

	== To know more ==		== To know more ==

Ros: /* Check out: key concepts and exercises */

2026-01-28T18:20:41Z

Check out: key concepts and exercises

← Older revision		Revision as of 20:20, 28 January 2026
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	== Check out: key concepts and exercises ==		== Check out: key concepts and exercises ==

	Freezing transition, entropy crisis, condensation, overlap distribution.		Freezing transition, typical vs average, domination by rare events, entropy crisis, condensation and extreme events, overlap distribution.

	After this lecture, you have all the tools to solve <code>Exercise 5 </code> and <code>Exercise 6 </code>		After this lecture, you have all the tools to solve <code>Exercise 5 </code> and <code>Exercise 6 </code>

Ros: /* Comments */

2026-01-28T18:19:58Z

Comments

← Older revision		Revision as of 20:19, 28 January 2026
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	== Comments==		== Comments==

			<ol>
			<li><em> Glassiness.</em> The low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also <em> a glass phase, </em> characterized by a non-trivial overlap distribution <math> P_\beta(q)</math> and by <math>q_{EA}=1.</math> These quantities can be considered as <em> order parameters </em> of the glass phase.<br></li>

	<em> Glassiness.</em> The low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also <em> a glass phase, </em> characterized by a non-trivial overlap distribution <math> P_\beta(q)</math> and by <math>q_{EA}=1.</math> These quantities can be considered as <em> order parameters </em> of the glass phase.<br>		<li><em> Replicas.</em> In the REM, <math> P_\beta(q)</math> can be computed explicitly using probability arguments; in more complicated models that require the machinery of replicas to compute the equilibrium properties, <math> P_\beta(q)</math> will emerge very naturally from such theory. We go back to this concepts in the next sets of problems, where we will also show that the non-zero order parameter <math> P_\beta(q)</math> is associated to a particular form of symmetry breaking, the so called replica symmetry.</li>
			</ol>
	<em> Replicas.</em> In the REM, <math> P_\beta(q)</math> can be computed explicitly using probability arguments; in more complicated models that require the machinery of replicas to compute the equilibrium properties, <math> P_\beta(q)</math> will emerge very naturally from such theory. We go back to this concepts in the next sets of problems, where we will also show that the non-zero order parameter <math> P_\beta(q)</math> is associated to a particular form of symmetry breaking, the so called replica symmetry.
	<br>

Ros: /* Comments */

2026-01-28T18:19:06Z

Comments

← Older revision		Revision as of 20:19, 28 January 2026
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	<em> Glassiness.</em> The low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also <em> a glass phase, </em> characterized by a non-trivial overlap distribution <math> P_\beta(q)</math> and by <math>q_{EA}=1.</math> These quantities can be considered as <em> order parameters </em> of the glass phase~~, associated to a particular form of symmetry breaking, the so called replica symmetry~~. In the REM, <math> P_\beta(q)</math> can be computed explicitly using probability arguments; in more complicated models that require the machinery of replicas to compute the equilibrium properties, <math> P_\beta(q)</math> will emerge very naturally from such theory. We go back to this concepts in the next sets of problems.		<em> Glassiness.</em> The low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also <em> a glass phase, </em> characterized by a non-trivial overlap distribution <math> P_\beta(q)</math> and by <math>q_{EA}=1.</math> These quantities can be considered as <em> order parameters </em> of the glass phase.<br>

			<em> Replicas.</em> In the REM, <math> P_\beta(q)</math> can be computed explicitly using probability arguments; in more complicated models that require the machinery of replicas to compute the equilibrium properties, <math> P_\beta(q)</math> will emerge very naturally from such theory. We go back to this concepts in the next sets of problems, where we will also show that the non-zero order parameter <math> P_\beta(q)</math> is associated to a particular form of symmetry breaking, the so called replica symmetry.
	<br>		<br>

Ros: /* Problem 2: the REM: freezing transition, condensation & glassiness */

2026-01-28T18:15:51Z

Problem 2: the REM: freezing transition, condensation & glassiness

← Older revision		Revision as of 20:15, 28 January 2026
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	<math>q(\vec{\sigma}^\alpha, \vec{\sigma}^\gamma)= \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \, \sigma_i^\beta</math>, and its distribution is		<math>q(\vec{\sigma}^\alpha, \vec{\sigma}^\gamma)= \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \, \sigma_i^\beta</math>, and its distribution is
	<math display="block">		<math display="block">
	P_{N, \beta}(q)= \sum_{\alpha, \~~beta~~} \frac{e^{-\beta E(\vec{\sigma}^\alpha)}}{Z_N}\frac{e^{-\beta E(\vec{\sigma}^\~~beta~~)}}{Z_N}\delta(q-q_{\alpha \~~beta~~}).		P_{N, \beta}(q)= \sum_{\alpha, \gamma} \frac{e^{-\beta E(\vec{\sigma}^\alpha)}}{Z_N(\beta)}\frac{e^{-\beta E(\vec{\sigma}^\gamma)}}{Z_N(\beta)}\delta(q-q(\vec{\sigma}^\alpha, \vec{\sigma}^\gamma)).
	</math>		</math>
	<br>		<br>

Ros: /* Problem 2: the REM: freezing transition, condensation & glassiness */

2026-01-28T18:15:07Z

Problem 2: the REM: freezing transition, condensation & glassiness

← Older revision		Revision as of 20:15, 28 January 2026
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	At <math> T_f </math> a "freezing transition" occurs: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the critical temperature <math>T= T_f </math>. <br>		At <math> T_f </math> a "freezing transition" occurs: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the critical temperature <math>T= T_f </math>. <br>
	We also characterize the overlap distribution of the model: the overlap between two configurations <math> \vec{\sigma}^\alpha, \vec{\sigma}^\~~beta~~ </math> is
	<math>q_{\alpha \, \~~beta~~}= \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \, \sigma_i^\beta</math>, and its distribution is		We also characterize the overlap distribution of the model: the overlap between two configurations <math> \vec{\sigma}^\alpha, \vec{\sigma}^\gamma </math> is
			<math>q(\vec{\sigma}^\alpha, \vec{\sigma}^\gamma)= \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \, \sigma_i^\beta</math>, and its distribution is
	<math display="block">		<math display="block">
	P_{N, \beta}(q)= \sum_{\alpha, \beta} \frac{e^{-\beta E(\vec{\sigma}^\alpha)}}{Z_N}\frac{e^{-\beta E(\vec{\sigma}^\beta)}}{Z_N}\delta(q-q_{\alpha \beta}).		P_{N, \beta}(q)= \sum_{\alpha, \beta} \frac{e^{-\beta E(\vec{\sigma}^\alpha)}}{Z_N}\frac{e^{-\beta E(\vec{\sigma}^\beta)}}{Z_N}\delta(q-q_{\alpha \beta}).

Ros: /* Check out: key concepts and exercises */

2026-01-21T20:51:32Z

Check out: key concepts and exercises

← Older revision		Revision as of 22:51, 21 January 2026
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	Freezing transition, entropy crisis, condensation, overlap distribution.		Freezing transition, entropy crisis, condensation, overlap distribution.

	After this lecture, you have all the tools to solve <code>Exercise X </code> and <code>Exercise Y </code>		After this lecture, you have all the tools to solve <code>Exercise 5 </code> and <code>Exercise 6 </code>

	== To know more ==		== To know more ==

Ros: /* To know more */

2026-01-20T12:29:16Z

To know more

← Older revision		Revision as of 14:29, 20 January 2026
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	== To know more ==		== To know more ==
	* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]		* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]
	* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]
	* Derrida and Toulouse. Sample to sample fluctuations in the random energy model [https://hal.science/jpa-00232503/document]		* Derrida and Toulouse. Sample to sample fluctuations in the random energy model [https://hal.science/jpa-00232503/document]
			<!--* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]-->

Ros: /* To know more */

2026-01-20T12:28:29Z

To know more

← Older revision		Revision as of 14:28, 20 January 2026
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	* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]		* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]
	* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]		* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]
	* Derrida et Toulouse. Sample to sample fluctuations in the random energy model [https://hal.science/jpa-00232503/document]		* Derrida and Toulouse. Sample to sample fluctuations in the random energy model [https://hal.science/jpa-00232503/document]

Ros: /* To know more */

2026-01-20T12:28:18Z

To know more

← Older revision		Revision as of 14:28, 20 January 2026
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	* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]		* Derrida. Random-energy model: limit of a family of disordered models [https://hal.science/hal-03285940v1/document]
	* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]		* Derrida. Random-energy model: An exactly solvable model of disordered systems [http://www.lps.ens.fr/~derrida/PAPIERS/1981/prb81.pdf]
	* [https://hal.science/jpa-00232503/document]		* Derrida et Toulouse. Sample to sample fluctuations in the random energy model [https://hal.science/jpa-00232503/document]