T-II-3 - Revision history

Ros: /* The order parameters: overlaps, and their meaning */

2023-12-22T11:25:26Z

The order parameters: overlaps, and their meaning

← Older revision		Revision as of 13:25, 22 December 2023
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	<center>		<center>
	<math>		<math>
	\overline{P}(q)=\lim_{n \to 0} \frac{2}{n(n-1)}\sum_{a>b}\delta \left(q- Q_{ab}^{SP}\right),		\overline{P}(q)=\lim_{n \to 0} \frac{2}{n(n-1)}\sum_{a>b}\delta \left(q- Q_{ab}^{SP}\right),\quad \quad \quad q_{EA}= \max \left\{ Q_{a \neq b}^{SP} \right\}
	</math>		</math>
	</center>		</center>

Ros: /* Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation */

2023-12-22T10:55:56Z

Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation

← Older revision		Revision as of 12:55, 22 December 2023
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	<br>		<br>

	<ol start=“3">		<ol start="3">
	<li>		<li>
	Compute the saddle point equations with respect to the parameter <math> q_0, q_1 </math> and <math> m </math> are. Check that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:		Compute the saddle point equations with respect to the parameter <math> q_0, q_1 </math> and <math> m </math> are. Check that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:

Ros: /* Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation */

2023-12-22T10:55:39Z

Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation

← Older revision		Revision as of 12:55, 22 December 2023
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	</math>		</math>
	</center>		</center>
	Here we have three parameters: <math>m, q_0, q_1</math> (in the sketch above, <math>m=3</math>).		Here we have three parameters: <math>m, q_0, q_1</math> (in the sketch above, <math>m=3</math>). We denote with <math>m^{SP}, q_0^{SP}, q_1^{SP}</math> their values at the saddle point.


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	<ol>		<ol>
			<li>
			What is the overlap distribution in this case?
			</li>
			</ol>
			<br>

			<ol start="2">
	<li>		<li>
	Using that		Using that
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	<br>		<br>

	<ol start="2">		<ol start=“3">
	<li>		<li>
	Compute the saddle point equations with respect to the parameter <math> q_0, q_1 </math> and <math> m </math> are. Check that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:		Compute the saddle point equations with respect to the parameter <math> q_0, q_1 </math> and <math> m </math> are. Check that <math> q_0=0</math> is again a valid solution of these equations, and that for <math> q_0=0</math> the remaining equations reduce to:
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	<br>		<br>

	<ol start="3">		<ol start="4">
	<li>		<li>
	We now look for a solution different from the paramagnetic one. To begin with, we set <math> m=1 </math> to satisfy the first equation, and look for a solution of		We now look for a solution different from the paramagnetic one. To begin with, we set <math> m=1 </math> to satisfy the first equation, and look for a solution of

Ros: /* Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation */

2023-12-22T10:54:01Z

Problem 3.2: the 1-RSB (Replica Symmetry Broken) calculation

← Older revision		Revision as of 12:54, 22 December 2023
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	</math>		</math>
	</center>		</center>
	Here we have three parameters: <math>m, q_0, q_1</math> (in the ~~formula~~ above, <math>m=3</math>).		Here we have three parameters: <math>m, q_0, q_1</math> (in the sketch above, <math>m=3</math>).

Ros: /* The order parameters: overlaps, and their meaning */

2023-12-22T10:53:20Z

The order parameters: overlaps, and their meaning

← Older revision		Revision as of 12:53, 22 December 2023
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	<center>		<center>
	<math>		<math>
	q_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\beta, \quad \quad \quad {P}(q)= \sum_{\alpha, \beta} \omega_\alpha\, \omega_\beta\, \delta(q- q_{\alpha \beta})		q_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\beta, \quad \quad \quad {P}(q)= \sum_{\alpha, \beta} \omega_\alpha\, \omega_\beta\, \delta(q- q_{\alpha \beta}).
	</math>		</math>
	</center>		</center>
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	</center>		</center>
	where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2. 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		where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2. 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}(q)</math>. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>.		non-trivial distribution <math> \overline{P}(q)</math>. This distribution measures the probability that two copies of the system, equilibrating in the sae disordered environment, end up having overlap <math>q</math>. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>.



	=== Problem 3.1: the RS (Replica Symmetric) calculation===		=== Problem 3.1: the RS (Replica Symmetric) calculation===

Ros: /* Problem 3.1: the RS (Replica Symmetric) calculation */

2023-12-22T10:49:44Z

Problem 3.1: the RS (Replica Symmetric) calculation

← Older revision		Revision as of 12:49, 22 December 2023
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	<br>		<br>

	<ol start=“3">		<ol start="3">
	<li>		<li>
	Compute the free energy corresponding to the solution <math>q_0= 0</math>, and show that it reproduces the annealed free energy. Do you have an interpretation for this?		Compute the free energy corresponding to the solution <math>q_0= 0</math>, and show that it reproduces the annealed free energy. Do you have an interpretation for this?

Ros: /* Problem 3.1: the RS (Replica Symmetric) calculation */

2023-12-22T10:49:25Z

Problem 3.1: the RS (Replica Symmetric) calculation

← Older revision		Revision as of 12:49, 22 December 2023
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	</math>		</math>
	</center>		</center>
	Under this assumption, there is a unique saddle point variable, that is <math>q_0</math>.		Under this assumption, there is a unique saddle point variable, that is <math>q_0</math>. We denote with <math>q_0^{SP}</math> its value at the saddle point.


	<ol>		<ol>
			<li>
			Under this assumption, what is the overlap distribution <math>\overline{P}(q)</math> and what is <math>q_{EA}</math>? In which sense the RS ansatz corresponds to assuming the existence of a unique pure state?
			</li>
			</ol>
			<br>

			<ol start="2">
	<li>		<li>
	Check that the inverse of the overlap matrix is		Check that the inverse of the overlap matrix is
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	<br>		<br>

	<ol start="2">		<ol start=“3">
	<li>		<li>
	Compute the free energy corresponding to the solution <math>q_0= 0</math>, and show that it reproduces the annealed free energy. Do you have an interpretation for this?		Compute the free energy corresponding to the solution <math>q_0= 0</math>, and show that it reproduces the annealed free energy. Do you have an interpretation for this?
	~~</li>~~
	~~</ol>~~
	~~<br>~~

	~~<ol start=“3">~~
	~~<li>~~
	~~Overlpa interpretation~~
	</li>		</li>
	</ol>		</ol>

Ros at 10:16, 22 December 2023

2023-12-22T10:16:42Z

← Older revision		Revision as of 12:16, 22 December 2023
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	<strong>Key concepts: </strong>		<strong>Key concepts: </strong>



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	where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2. 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		where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2. 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}(q)</math>. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>.		non-trivial distribution <math> \overline{P}(q)</math>. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>.



	=== Problem 3.1: the RS (Replica Symmetric) calculation===		=== Problem 3.1: the RS (Replica Symmetric) calculation===

	~~<!-- Remember however that~~ to ~~get~~ the ~~free energy, we are interested in~~ the ~~limit~~ <math>~~n \to 0~~</math>. A standard way to proceed is: (i) make an ansatz on the structure of the matrix Q, (ii) compute <math>\mathcal{A}[Q]</math> within this ansatz and expand <math>\mathcal{A}[Q]= n\mathcal{A}_0 + O(n^2)</math>, (iii) perform the ~~saddle-point calculation on <math>\mathcal{A}_0</math>~~. ~~-->~~		We go back to the saddle point equations for the spherical <math>p</math>-spin model derived in the previous problems. Let us consider the simplest possible ansatz for the structure of the matrix Q, that is the Replica Symmetric (RS) ansatz:

	Let us consider the simplest possible ansatz for the structure of the matrix Q, that is the Replica Symmetric (RS) ansatz:

	<center>		<center>

Ros: /* The order parameters: overlaps, and their meaning */

2023-12-22T10:14:47Z

The order parameters: overlaps, and their meaning

← Older revision		Revision as of 12:14, 22 December 2023
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	</math>		</math>
	</center>		</center>
	In the Ising model at low temperature there are two pure states, that correspond to positive and negative magnetization. In a mean-field spin glass, there are more than two pure states. The quantity <math>q_{EA}</math> measures the overlap between configurations belonging to the same pure state, that one expects to be the same for all states:		In the Ising model at low temperature there are two pure states, <math>\alpha= \pm 1 </math>, that correspond to positive and negative magnetization. In a mean-field spin glass, there are more than two pure states. The quantity <math>q_{EA}</math> measures the overlap between configurations belonging to the same pure state, that one expects to be the same for all states:
	<center>		<center>
	<math>		<math>
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	</math>		</math>
	</center>		</center>
	where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2.		where <math> Q_{ab}^{SP}</math> are the saddle point values of the overlap matrix introduced in Problem 2.2. 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}(q)</math>. In the Ising case, a low temperature one has <math> q_{\alpha \alpha}=m^2</math> and <math> q_{\alpha \neq \beta}=-m^2</math>, and thus <math> \overline{P}(q)</math> has two peaks at <math> \pm m^2</math>.
	The solution of the saddle point equations for the overlap matrix <math> Q</math> thus ~~encodes a lot of information on the system; in particular, they~~ contain the information on whether spin glass order emerges:

	~~where <math> \overline{P}(q)</math> is the probability that two copies of the system~~, ~~equilibrating in the same disordered environment, end up~~ having ~~overlap q with each others. A~~ non-trivial distribution <math> \overline{P}(q)</math> ~~is an indicator of ergodicity breaking in the systems: it indicates that the Boltzmann measure is partitioned into several <em>pure states</em> (like~~ the <math> +m</math> and <math> -m</math> ~~states in the Ising case, where <math> m</math> is the magnetization)~~, and ~~it measures the typical overlaps between configurations belonging to the same, or different states. In Ising,~~ <math> \overline{P}(q)= \~~delta(q-~~m^2)</math>.

	=== Problem 3.1: the RS (Replica Symmetric) calculation===		=== Problem 3.1: the RS (Replica Symmetric) calculation===

Ros: /* The order parameters: overlaps, and their meaning */

2023-12-22T09:53:18Z

The order parameters: overlaps, and their meaning

← Older revision		Revision as of 11:53, 22 December 2023
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	<center>		<center>
	<math>		<math>
	q_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\beta, \quad \quad \quad {P}(q)= \sum_{\alpha, \beta} \~~frac{Z_~~\alpha~~}{Z}~~\, \~~frac{Z_~~\beta~~}{Z}~~\, \delta(q- q_{\alpha \beta})		q_{\alpha \beta}= \frac{1}{N}\sum_i \langle \sigma_i \rangle_\alpha \langle \sigma_i \rangle_\beta, \quad \quad \quad {P}(q)= \sum_{\alpha, \beta} \omega_\alpha\, \omega_\beta\, \delta(q- q_{\alpha \beta})
	</math>		</math>
	</center>		</center>