L-8 - Revision history

Rosso: /* Verification of Furstenberg's hypotheses for the Anderson model */

2026-03-23T11:36:36Z

Verification of Furstenberg's hypotheses for the Anderson model

← Older revision		Revision as of 13:36, 23 March 2026
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	=== (i) Stretching ===		=== (i) Stretching ===

	We first verify that ~~each matrix expands~~ at least one direction.		We first verify that the matrices expand at least one direction with nonzero probability.

	For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here		For a generic <math>2\times2</math> matrix, the maximal stretching factor is controlled by the largest eigenvalue of <math>T_n^T T_n</math>. Here
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	</math>		</math>

	Thus ~~each matrix expands~~ at least one direction, except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.		Thus the matrices expand at least one direction except for the fine-tuned case <math>V_n=\epsilon</math>, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

	=== (ii) Mixing of directions ===		=== (ii) Mixing of directions ===
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	For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.		For fixed <math>\theta_n</math>, this expression depends continuously on the random variable <math>V_n</math>. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

	As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions: under ~~iteration~~ of ~~the product~~		As a consequence, the sequence <math>\theta_n</math> is not confined to a finite set of directions. More precisely, there is no finite collection of directions that is invariant under all transfer matrices.
	<math ~~display="block"~~>
	~~T_n T_{n-1} \cdots T_1,~~		The randomness of <math>V_n</math> continuously reshuffles the direction and prevents the dynamics from being trapped into special directions. This ensures that the projective dynamics effectively explores the angular space.
	</math>
	the ~~angular~~ dynamics ~~can access a continuum of~~ directions.

	This ~~exploration~~ is ~~not required to be uniform; it~~ is ~~sufficient that all~~ directions ~~are accessible. This is precisely~~ the ~~mixing condition required by Furstenberg's theorem~~.		This is what is meant by '''mixing of directions''' in the present context.

	=== Conclusion ===		=== Conclusion ===
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	The transfer matrices of the one-dimensional Anderson model satisfy both conditions:		The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

	# they expand at least one direction;		# they expand at least one direction with nonzero probability;
	# they do not confine the angular dynamics to a finite set.		# they do not confine the angular dynamics to a finite set.

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	</math>		</math>
	so the Lyapunov exponent is positive for generic disorder.		so the Lyapunov exponent is positive for generic disorder.




	---		---

Rosso: /* Furstenberg theorem (physical formulation) */

2026-03-23T11:34:14Z

Furstenberg theorem (physical formulation)

← Older revision		Revision as of 13:34, 23 March 2026
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	\Pi_n = T_n T_{n-1} \cdots T_1		\Pi_n = T_n T_{n-1} \cdots T_1
	</math>		</math>
	can be understood as a matrix analogue of the law of large numbers. A sufficient ~~condition~~ for this behavior is provided by a theorem of Harry Furstenberg.		can be understood as a matrix analogue of the law of large numbers. A set of simple sufficient conditions for this behavior is provided by a theorem of Harry Furstenberg.

	To quantify this growth, we introduce the norm of a matrix <math>A</math> as		To quantify this growth, we introduce the norm of a matrix <math>A</math> as
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	</math>		</math>

	In physical terms, the theorem relies on two key ingredients:		In physical terms, the mechanism behind the theorem relies on two key ingredients:

	=== (i) Stretching ===		=== (i) Stretching ===

	~~Each matrix~~ must be able to expand ~~at least one direction~~. ~~This means that for each~~ <math>n</math> ~~there exists a vector~~ <math>v</math> ~~such that~~		The transfer matrices must be able, with nonzero probability, to expand vectors.

			More precisely, there exist realizations of <math>T_n</math> such that for some direction <math>v</math>,
	<math display="block">		<math display="block">
	\\|T_n v\\| = \sigma_{\max}(T_n)\, \\|v\\|,		\\|T_n v\\| = \sigma_{\max}(T_n)\, \\|v\\|,
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	</math>		</math>

	Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of ~~the symmetric matrix~~		Equivalently, <math>\sigma_{\max}^2(T_n)</math> is the largest eigenvalue of
	<math display="block">		<math display="block">
	T_n^T T_n.		T_n^T T_n.
	</math>		</math>

			This ensures that typical products contain episodes of exponential amplification.

	=== (ii) Mixing of directions ===		=== (ii) Mixing of directions ===
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	\theta_{n+1}=F(\theta_n).		\theta_{n+1}=F(\theta_n).
	</math>		</math>
	Once the initial direction is fixed, the whole sequence is fixed. The ~~direction may change, but it keeps~~ a perfect memory of its initial ~~value~~.		Once the initial direction is fixed, the whole sequence is fixed. The system retains a perfect memory of its initial orientation.

	In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes		In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes
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	\theta_{n+1}=F(\theta_n,T_n).		\theta_{n+1}=F(\theta_n,T_n).
	</math>		</math>
	The direction is then continuously reshuffled and progressively loses memory of its initial ~~orientation~~.		The direction is then continuously reshuffled and progressively loses memory of its initial value.

	More ~~precisely~~, there is no finite set of directions that is invariant under all transfer matrices. ~~The~~ dynamics ~~therefore cannot be reduced to a finite set of preferred~~ directions.		More importantly, there is no finite set of directions that is invariant under all transfer matrices. As a consequence, the dynamics does not get trapped into special directions and effectively explores the projective space.

	This is what is meant by '''mixing of directions'''~~, and it is the key condition required by Furstenberg's theorem~~.		This absence of invariant directions is what is meant by '''mixing of directions'''.

	=== Consequence ===		=== Consequence ===
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	where <math>\gamma</math> is the Lyapunov exponent.		where <math>\gamma</math> is the Lyapunov exponent.

	This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (~~see Exercise~~ 15).		This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (exercise 15).


	== Verification of Furstenberg's hypotheses for the Anderson model ==		== Verification of Furstenberg's hypotheses for the Anderson model ==

Rosso: /* (ii) Mixing of directions */

2026-03-23T06:16:40Z

(ii) Mixing of directions

← Older revision		Revision as of 08:16, 23 March 2026
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	=== (ii) Mixing of directions ===		=== (ii) Mixing of directions ===

	~~The~~ angular dynamics is ~~not confined to a finite set of directions: under iteration of the product~~		At each step, the transfer matrix changes the direction of the vector <math>(\psi_n,\psi_{n-1})</math>.

			In the clean case, the same matrix is applied at every step, so the angular dynamics is deterministic and can be written as
	<math display="block">		<math display="block">
	~~T_n T_~~{n-1} \~~cdots T_1,~~		\theta_{n+1}=F(\theta_n).
	</math>		</math>
	the direction of a ~~vector does not preserve any~~ finite set of directions		Once the initial direction is fixed, the whole sequence is fixed. The direction may change, but it keeps a perfect memory of its initial value.

			In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes
			<math display="block">
			\theta_{n+1}=F(\theta_n,T_n).
			</math>
			The direction is then continuously reshuffled and progressively loses memory of its initial orientation.

			More precisely, there is no finite set of directions that is invariant under all transfer matrices. The dynamics therefore cannot be reduced to a finite set of preferred directions.

			This is what is meant by '''mixing of directions''', and it is the key condition required by Furstenberg's theorem.

	=== Consequence ===		=== Consequence ===

Rosso: /* (ii) Mixing of directions */

2026-03-22T19:10:26Z

(ii) Mixing of directions

← Older revision		Revision as of 21:10, 22 March 2026
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	=== (ii) Mixing of directions ===		=== (ii) Mixing of directions ===

	The angular dynamics is not confined: under iteration of the product		The angular dynamics is not confined to a finite set of directions: under iteration of the product
	<math display="block">		<math display="block">
	T_n T_{n-1} \cdots T_1,		T_n T_{n-1} \cdots T_1,
	</math>		</math>
	the direction of a vector ~~can explore the whole angular space. This exploration is~~ not ~~required to be uniform; it is sufficient that all~~ directions ~~are accessible.~~		the direction of a vector does not preserve any finite set of directions

	=== Consequence ===		=== Consequence ===

Rosso: /* Density of states */

2026-03-22T15:03:57Z

Density of states

← Older revision		Revision as of 17:03, 22 March 2026
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	In the presence of disorder the density of states broadens and becomes sample dependent.		In the presence of disorder the density of states broadens and becomes sample dependent.

	~~---~~

	== Transfer matrices ==		== Transfer matrices ==

Rosso: /* Transfer matrices */

2026-03-21T15:58:06Z

Transfer matrices

← Older revision		Revision as of 17:58, 21 March 2026
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	== Transfer matrices ==		== Transfer matrices ==

	~~After multiplying the eigenvalue equation by <math>-1</math> and redefining~~		The discrete Schrödinger equation reads
	~~<math>-V_n \to V_n</math> and <math>-\epsilon \to \epsilon</math>, the~~ discrete Schrödinger equation reads

	<math display="block">		<math display="block">
	\psi_{n+1} + \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .		-\psi_{n+1} - \psi_{n-1} + V_n \psi_n = \epsilon \psi_n .
	</math>		</math>

Rosso: /* Anderson model (tight-binding model) */

2026-03-21T15:56:05Z

Anderson model (tight-binding model)

← Older revision		Revision as of 17:56, 21 March 2026
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	</math>		</math>

	The random variables <math>V_i</math> represent on-site disorder.		The random variables <math>V_i</math> represent on-site disorder.
			For simplicity we set <math>t=1</math>.

	~~For simplicity we set~~		We assume that the disorder variables are independent and identically distributed, uniformly in the interval

	~~<math>t=1</math>.~~

	~~The~~ disorder variables are independent ~~random variables drawn from~~ the ~~box distribution~~

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

	In one dimension the single-particle Hamiltonian ~~corresponds to a~~ tridiagonal matrix		In one dimension, the single-particle Hamiltonian is represented by the tridiagonal matrix

	<math display="block">		<math display="block">
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	\sum_{n=1}^L \|\psi_n\|^2 = 1 .		\sum_{n=1}^L \|\psi_n\|^2 = 1 .
	</math>		</math>

			In the one-particle sector, this becomes a discrete Schrödinger equation.

	---		---
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	== Transfer matrices ==		== Transfer matrices ==

	~~The~~ discrete Schrödinger equation reads		After multiplying the eigenvalue equation by <math>-1</math> and redefining
			<math>-V_n \to V_n</math> and <math>-\epsilon \to \epsilon</math>, the discrete Schrödinger equation reads

	<math display="block">		<math display="block">
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	Thus the wavefunction is controlled by a product of random matrices.		Thus the wavefunction is controlled by a product of random matrices.

	~~---~~		=== Remark ===

			The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

			This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

	== Furstenberg theorem (physical formulation) ==		== Furstenberg theorem (physical formulation) ==

Rosso: /* Lyapunov exponent */

2026-03-20T22:37:42Z

Lyapunov exponent

Show changes

Rosso at 20:39, 12 March 2026

2026-03-12T20:39:21Z

Show changes

Rosso at 17:44, 3 March 2026

2026-03-03T17:44:42Z

← Older revision		Revision as of 19:44, 3 March 2026
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	==Transfer matrices and Lyapunov exponents==		==Transfer matrices and Lyapunov exponents==

	~~=== Product of random variables and Central limit theorem ===~~

	~~Consider a set of positive iid random variables <math>x_1,x_2,\ldots x_N</math> with finite mean and variance and compute their product~~
	~~<center><math>~~
	~~\Pi_N= \prod_{n=1}^N x_i, \quad \text{or} \; \ln \Pi_N= \sum_{n=1}^N \ln x_i~~
	~~</math></center>~~
	~~For large N, the Central Limit Theorem predicts:~~
	~~<center><math>~~
	~~\log \Pi_N= \gamma N + \gamma_2 \sqrt{N} \chi~~
	~~</math></center>~~
	* <math>\chi</math> is a Gaussian number of zero mean and unit variance
	* <math>\gamma, \gamma_2</math> are N indepent and can be written as
	~~<center><math>~~
	~~\gamma =\overline{\ln x}< \ln \overline{x}, \quad \gamma_2= \sqrt{\overline{(\ln x)^2}-(\overline{\ln x})^2}~~
	~~</math></center>~~

	~~=== Log-normal distribution ===~~
	~~The distribution of <math>\Pi_N</math> is log-normal~~
	~~<center><math>~~
	~~P(\Pi_N) d \Pi_N = \frac{1}{ \sqrt{2 \pi \gamma_2^2 N}} \exp\left[-\frac{(\ln(\Pi_N)-\gamma N)^2}{2 \gamma_2^2 N}\right] \frac{d\Pi_N}{\Pi_N}~~
	~~</math></center>~~

	~~<Strong> Quenched and Annealed averages </Strong>~~

	~~To compute the moments of the log-normal distribution, it is convenient to introduce the variable~~
	~~<center><math> X \equiv \ln(\Pi_N) </math></center> which is Gaussian distributed:~~

	~~<center><math> p(X) = \frac{1}{ \sqrt{2 \pi \sigma^2}} \exp\left[-\frac{(X-\mu)^2}{2 \sigma^2}\right] </math></center>~~
	~~with <math>\mu =\gamma N</math> and <math>\sigma^2=\gamma_2^2 N</math>.~~

	The moments of <math>\Pi_N</math> can be easily computed: <center><math>\overline{\Pi_N^n} = \int dX \, e^{nX} p(X) = \exp\left[\mu n +\sigma^2 \frac{n^2}{2} \right]=\exp\left[(\gamma n +\gamma_2^2 \frac{n^2}{2})N \right] </math> </center>

	~~The variable <math>\Pi_N</math> is therefore not self-averaging (see Valentina's lecture 1) since its fluctuations grow with <math>N</math> faster than its mean:~~
	~~<center><math> \frac{\overline{\Pi_N^2}}{(\overline{\Pi_N})^2}= \exp\left[\gamma_2^2 N \right] </math></center>~~

	~~Hence, <math>\Pi_N</math> is not self-averaging, while <math>\ln \Pi_N</math> is self-averaging.~~

	~~In particular, the mean <math>\overline{\Pi_N} = \exp[(\gamma+\gamma_2^2/2) N]</math> grows much faster than the typical value <math>\Pi_N^{\text{typ}} \equiv \exp(\gamma N)</math>.~~

	== Product of random matrices==		== Product of random matrices==