L-4 - Revision history

Rosso at 20:05, 1 March 2026

2026-03-01T20:05:01Z

Rosso: /* Second Moment */

2026-02-02T10:47:21Z

Second Moment

← Older revision		Revision as of 12:47, 2 February 2026
Line 144:		Line 144:
	Z_{\text{free}}(u=0,t,2T),		Z_{\text{free}}(u=0,t,2T),
	\qquad		\qquad
	Z_{\text{free}}(u=0,t,2T) = (4\pi T t)^{N/2}.		Z_{\text{free}}(u=0,t,2T) = (4\pi T t)^{-N/2}.
	</math>		</math>
	</center>		</center>

Rosso: /* State of the Art */

2026-02-02T09:48:18Z

State of the Art

← Older revision		Revision as of 11:48, 2 February 2026
Line 9:		Line 9:

	* In <math>N=1</math>, one has <math>\theta=1/3</math> and a glassy regime present at all temperatures.		* In <math>N=1</math>, one has <math>\theta=1/3</math> and a glassy regime present at all temperatures.
	The model is integrable through a non-standard Bethe Ansatz, and the distribution of <math>E_{\min}</math> for a given boundary condition is of the Tracy–Widom type.		The model is integrable through a non-standard Bethe Ansatz, and the distribution of <math>E_{\min}</math> for a given boundary condition is of the Tracy–Widom type.

	* In <math>N=\infty</math>, corresponding to the Cayley tree, an exact solution exists, predicting a freezing transition to a one-step replica symmetry breaking phase (<math>\theta=0</math>).		* In <math>N=\infty</math>, corresponding to the Cayley tree, an exact solution exists, predicting a freezing transition to a one-step replica symmetry breaking phase (<math>\theta=0</math>).

Rosso: /* Directed Polymer in finite dimension */

2026-02-02T09:48:04Z

Directed Polymer in finite dimension

Show changes

Rosso: /* Back to REM: condensation of the Gibbs measure */

2026-02-02T09:45:07Z

Back to REM: condensation of the Gibbs measure

Show changes

Rosso: /* Let's do replica! */

2026-02-01T20:44:29Z

Let's do replica!

← Older revision		Revision as of 22:44, 1 February 2026
Line 35:		Line 35:
	<center> <math> \overline{V(x,\tau)} = 0, \qquad \overline{V(x,\tau) V(x',\tau')} = D \, \delta^d(x-x') \, \delta(\tau - \tau'). </math> </center>		<center> <math> \overline{V(x,\tau)} = 0, \qquad \overline{V(x,\tau) V(x',\tau')} = D \, \delta^d(x-x') \, \delta(\tau - \tau'). </math> </center>
	* Since the disorder is Gaussian, averages of exponentials can be computed using Wick’s theorem:		* Since the disorder is Gaussian, averages of exponentials can be computed using Wick’s theorem:
	<center> <math> \overline{\exp(W)} = \exp\!\Big[\overline{W} + \frac{1}{2}\big(\overline{W^2} - \overline{W}^2\big)\Big], </math> </center>		<center> <math> \overline{\exp(W)} = \exp\!\Big[\overline{W} + \frac{1}{2}\big(\overline{W^2} - (\overline{W})^2\big)\Big], </math> </center>
	for any Gaussian random variable <math>W</math>.		for any Gaussian random variable <math>W</math>.

Rosso: /* Back to REM: condensation of the Gibbs measure */

2025-09-16T18:14:49Z

Back to REM: condensation of the Gibbs measure

← Older revision		Revision as of 20:14, 16 September 2025
Line 102:		Line 102:
	=== Behavior in Different Phases:===		=== Behavior in Different Phases:===
	* '''High-Temperature Phase (<math> \beta < \beta_c = 1/b_M = \sqrt{2 \log2}</math>):'''		* '''High-Temperature Phase (<math> \beta < \beta_c = 1/b_M = \sqrt{2 \log2}</math>):'''
	: In this regime, the total weight of the excited states dominates over the weight of the ground state. The ground state is therefore not deep enough to overcome the finite entropy contribution. As a result, the probability of sampling the same configuration twice from the Gibbs measure is exponentially small in ~~'N'~~.		: In this regime, the total weight of the excited states dominates over the weight of the ground state. The ground state is therefore not deep enough to overcome the finite entropy contribution. As a result, the probability of sampling the same configuration twice from the Gibbs measure is exponentially small in the system size.


Line 141:		Line 141:
	When <math>\theta > 0</math>, one must extend the definition of this exponent to finite temperatures and consider the fluctuations of the free energy <math>F(L,\beta)</math>. Three representative cases are:		When <math>\theta > 0</math>, one must extend the definition of this exponent to finite temperatures and consider the fluctuations of the free energy <math>F(L,\beta)</math>. Three representative cases are:

	* '''Directed polymer in <math>D=1,2</math>:'''		* '''Directed polymer in <math>N=1,2</math>:'''
	The fluctuations of the ground state exhibit a positive, temperature-independent exponent <math>\theta</math>. In this situation, only the glassy phase exists, and		The fluctuations of the ground state exhibit a positive, temperature-independent exponent <math>\theta</math>. In this situation, only the glassy phase exists, and

	<center><math>P(q) = \delta(1-q),</math></center> because producing an excitation with vanishing overlap with the ground state is very costly.		<center><math>P(q) = \delta(1-q),</math></center> because producing an excitation with vanishing overlap with the ground state is very costly.

	* '''Directed polymer in <math>D=3</math>:'''		* '''Directed polymer in <math>N=3</math>:'''
	The exponent <math>\theta</math> depends on the temperature: it vanishes above the glass transition and becomes strictly positive below it. Accordingly,		The exponent <math>\theta</math> depends on the temperature: it vanishes above the glass transition and becomes strictly positive below it. Accordingly,

Rosso: /* Overlap Distribution and Replica Symmetry Breaking: */

2025-09-16T13:18:19Z

Overlap Distribution and Replica Symmetry Breaking:

← Older revision		Revision as of 15:18, 16 September 2025
Line 129:		Line 129:
	<center> <math>P(q) = \tfrac{\beta_c}{\beta}\,\delta(q) + \Bigl(1 - \tfrac{\beta_c}{\beta}\Bigr)\,\delta(1-q).</math> </center>		<center> <math>P(q) = \tfrac{\beta_c}{\beta}\,\delta(q) + \Bigl(1 - \tfrac{\beta_c}{\beta}\Bigr)\,\delta(1-q).</math> </center>


			== Finite Dimensional Systems==
	In finite dimensions, the fluctuations of the ground-state energy are characterized by an exponent <math>\theta</math>:		In finite dimensions, the fluctuations of the ground-state energy are characterized by an exponent <math>\theta</math>:

Line 152:		Line 154:
	The behavior is analogous to the Random Energy Model: <math>\theta = 0</math> in both phases. At high temperature,		The behavior is analogous to the Random Energy Model: <math>\theta = 0</math> in both phases. At high temperature,

	<center><math>P(q) = \delta(q),</math></center> while at low temperature the system exhibits the one-step replica symmetry breaking picture.		<center><math>P(q) = \delta(q),</math></center> while at low temperature the system exhibits the one-step replica symmetry breaking picture:

			<center> <math>P(q) = \tfrac{\beta_c}{\beta}\,\delta(q) + \Bigl(1 - \tfrac{\beta_c}{\beta}\Bigr)\,\delta(1-q).</math> </center>

Rosso: /* More general REM and systems in Finite dimensions */

2025-09-16T13:04:43Z

More general REM and systems in Finite dimensions

← Older revision		Revision as of 15:04, 16 September 2025
Line 129:		Line 129:
	<center> <math>P(q) = \tfrac{\beta_c}{\beta}\,\delta(q) + \Bigl(1 - \tfrac{\beta_c}{\beta}\Bigr)\,\delta(1-q).</math> </center>		<center> <math>P(q) = \tfrac{\beta_c}{\beta}\,\delta(q) + \Bigl(1 - \tfrac{\beta_c}{\beta}\Bigr)\,\delta(1-q).</math> </center>

	~~==More general REM and systems in Finite~~ dimensions==		In finite dimensions, the fluctuations of the ground-state energy are characterized by an exponent <math>\theta</math>:

	In random energy models with i.i.d. random variables, the distribution <math>p(E)</math> determines the dependence of <math>a_M</math> and <math>b_M</math> on ''M'', and consequently their scaling with ''N'', the number of degrees of freedom. It is insightful to consider a more general case where an exponent <math>\omega</math> describes the fluctuations of the ground state energy:		<center> <math>\overline{\big(E_{\min} - \overline{E_{\min}}\big)^2} \sim L^{2\theta},</math> </center>
	<center> <math>\overline{\~~left~~(E_{\min} - \overline{E_{\min}}\~~right~~)^2} \sim ~~b_M^2 \propto N~~^{2\~~omega~~}</math> </center>

	~~Three distinct scenarios emerge depending on~~ the ~~sign~~ of <math>~~\omega~~</math>:		where <math>L</math> is the linear size of the system and <math>N = L^D</math> is the number of degrees of freedom.

	* For <math>\~~omega~~ < 0</math>~~: The freezing~~ temperature vanishes with increasing system size, leading to the absence of a ~~freezing~~ transition. This scenario occurs in many low-dimensional systems, such as the ~~Edwards-Anderson~~ model in two dimensions.		When <math>\theta < 0</math>, the critical temperature vanishes with increasing system size, leading to the absence of a glass transition. This scenario occurs in many low-dimensional systems, such as the Edwards–Anderson model in two dimensions.

	* For <math>\~~omega =~~ 0</math>~~: A freezing transition is guaranteed. For~~ the ~~Random Energy Model discussed earlier,~~ <math>~~T_f = 1/~~\~~sqrt{2 \log 2}~~</math>. An important feature of this transition, as will be explored in the next section, is that condensation does not occur solely in the ground state but also involves a large (albeit not extensive) number of low-energy excitations.		When <math>\theta > 0</math>, one must extend the definition of this exponent to finite temperatures and consider the fluctuations of the free energy <math>F(L,\beta)</math>. Three representative cases are:

	* ~~For~~ <math>~~\omega > 0~~</math>: The ~~freezing temperature grows with the system size, resulting in only the glassy phase. The system condenses entirely into~~ the ground state ~~since the excited states are characterized by prohibitively high energies. This scenario~~, ~~while less intricate than the~~ <math>\~~omega = 0~~</math> ~~case~~, ~~corresponds to a~~ glassy phase ~~with a single deep ground state.~~		* '''Directed polymer in <math>D=1,2</math>:'''
			The fluctuations of the ground state exhibit a positive, temperature-independent exponent <math>\theta</math>. In this situation, only the glassy phase exists, and

	The extent to which these scenarios persist in real systems in finite dimensions remains an open question. In these systems, such as the directed polymers, the fluctuations of the ground state energy are characterized by an exponent <math>\theta</math>:		<center><math>P(q) = \delta(1-q),</math></center> because producing an excitation with vanishing overlap with the ground state is very costly.
	<center> <math>\~~overline{\left~~(~~E_{\min}~~ - ~~\overline{E_{\min}}\right~~)~~^2} \sim L^{2\theta}~~</math> </center>

	~~where~~ <math>L</math> is the ~~linear size of~~ the ~~system~~ and <math>N = ~~L^D~~</math> ~~is the number of degrees of freedom~~.		* '''Directed polymer in <math>D=3</math>:'''
			The exponent <math>\theta</math> depends on the temperature: it vanishes above the glass transition and becomes strictly positive below it. Accordingly,

			<center><math>P(q) = \delta(1-q)</math> at low temperature, and <math>P(q) = \delta(q)</math> at high temperature.</center>

	~~At finite temperatures, an analogous exponent is defined by studying the fluctuations of the free energy, <math>F = E - T S</math>. For the directed~~ polymer ~~in low dimwnsion~~ the ~~fluctuations of~~ the ~~ground state exhibit a positive and temperature-independent~~ <math>\theta~~</math>. In such cases, only the glassy phase exists, aligning with the <math>\omega >~~ 0</math> ~~scenario~~ in ~~REMs~~.		* '''Directed polymer on the Cayley tree:'''
			The behavior is analogous to the Random Energy Model: <math>\theta = 0</math> in both phases. At high temperature,

	~~On the other hand, for the directed polymer in high dimension,~~ <math>\~~theta~~</math> ~~is positive~~ at low ~~temperatures but vanishes at high temperatures. This behavior defines a glass transition mechanism entirely distinct from those observed in mean~~-~~field models~~.		<center><math>P(q) = \delta(q),</math></center> while at low temperature the system exhibits the one-step replica symmetry breaking picture.

Rosso: /* Overlap Distribution and Replica Symmetry Breaking: */

2025-09-16T12:33:02Z

Overlap Distribution and Replica Symmetry Breaking:

← Older revision		Revision as of 14:33, 16 September 2025
Line 115:		Line 115:

	====Overlap Distribution and Replica Symmetry Breaking:====		====Overlap Distribution and Replica Symmetry Breaking:====
	The structure of states can be further characterized through the overlap between two configurations <math>\alpha</math> and <math>\~~beta~~</math>, defined as		The structure of states can be further characterized through the overlap between two configurations <math>\alpha</math> and <math>\gamma</math>, defined as

	<center> <math> q_{\alpha,\~~beta~~} = \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \sigma_i^\~~beta~~, </math> </center>		<center> <math> q_{\alpha,\gamma} = \frac{1}{N} \sum_{i=1}^N \sigma_i^\alpha \sigma_i^\gamma, </math> </center>

	which takes values in the interval <math>(-1,1)</math>. The distribution <math>P(q)</math> of the overlap between two configurations sampled from the Gibbs measure distinguishes the two phases:		which takes values in the interval <math>(-1,1)</math>. The distribution <math>P(q)</math> of the overlap between two configurations sampled from the Gibbs measure distinguishes the two phases: