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<!--<strong>Goal:</strong>  in this final set of problems, we discuss the interplay between localization and glassiness, by connecting the solution to the Anderson problem on the Bethe lattice with the statistical physics problem of a directed polymer in random media on trees.
 
 
 
 
<strong>Goal:</strong>  in this final set of problems, we discuss the interplay between localization and glassiness, by connecting the solution to the Anderson problem on the Bethe lattice with the statistical physics problem of a directed polymer in random media on trees.
<br>
<br>
<strong>Techniques: </strong>   
<strong>Techniques: </strong>   
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=== Problem 7.2:  localization-delocalization transition on the Bethe lattice ===
We now focus on the self energies, since the criterion for localization is given in terms of these quantities. In this Problem we will determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs.
<ol>
<li><em> The “localized" solution. </em> We set <math> z=E+ i \eta </math> and <math> \sigma^{\text{cav}}_{a}(z)= R_a(z) -i \Gamma_a(z)</math>. Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:
<center>
<math>
\Gamma_a= \sum_{b \in \partial a} t_{ab}^2 \frac{\Gamma_b + \eta}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2}, \quad \quad R_a =  \sum_{b \in \partial a} t_{ab}^2 \frac{E- \epsilon_b - R_b}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2}
</math>
</center>
Justify why the solution corresponding to localization, <math> \Gamma_a=0 </math>, is always a solution when <math> \eta \to 0 </math>; moreover, in the localized phase when <math> \eta </math> is finite but small one has <math> \Gamma_a \sim O(\eta) </math>. How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?
</li><br>
<li><em> Imaginary approximation and distributional equation. </em> We consider the equations for <math> \Gamma_a </math> and neglect the terms <math> R_b </math> in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where <math> \Gamma_a \sim \eta \ll 1 </math>. Finally, we set <math> t_{ab} \equiv t </math> and <math> E=0 </math> for simplicity. Show that under these assumptions the probability density for the imaginary part, <math> P_\Gamma(\Gamma)</math>, satisfies
<center>
<math>
P_\Gamma(\Gamma)= \int \prod_{b=1}^k d\epsilon_b\,p(\epsilon_b)\int  \prod_{c=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - t^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ \epsilon_b^2}  \right)
</math>
</center>
Show that the Laplace transform of this distribution, <math> \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) </math>, satisfies
<center>
<math>
\Phi(s)= \left[ \int d\epsilon\, p(\epsilon) e^{-\frac{s t^2 \eta}{\epsilon^2}} \Phi \left(\frac{s t^2 }{\epsilon^2} \right)  \right]^k
</math>
</center>
</li><br>
<li><em> The stability analysis. </em> We now wish to check the stability of our assumption to be in the localized phase,  <math> \Gamma_a \sim \eta \ll 1 </math>, which led to the identity above for <math> \Phi(s) </math>. Our assumption is that the typical value of <math> \Gamma_a </math> is small, except for cases in which one of the descendants <math> b </math> is such that <math> \epsilon_b </math> is very small, in which case <math> \Gamma_a \sim 1/ \epsilon_b^2 </math>.
<ul>
<li> Show that if <math> \Gamma \sim 1/ \epsilon^2 </math> and <math>p(\epsilon)</math> is not gapped around zero, then <math>P_\Gamma(\Gamma) \sim \Gamma^{-3/2}</math>, i.e. the distribution has tails contributed by these events in which the local fields happen to be very small.  </li>
<li> Assume more generally that  <math>P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}</math> for large <math> \Gamma </math> and <math> \alpha \in [1, 3/2]</math>. Show that this corresponds to <math> \Phi(s) \sim 1- A |s|^\beta </math> for <math> s </math> small, with <math> \beta= \alpha-1 \in [0, 1/2] </math>.  </li>
<li>  Show that the equation for <math> \Phi(s) </math> gives for <math> s </math> small <math>1- A s^\beta =1- A k \int d\epsilon \, p(\epsilon) \frac{s^\beta t^{2 \beta}}{\epsilon^{2 \beta}}+ o(s^\beta) </math>, and therefore this is consistent provided that there exists a <math> \beta \in [0, 1/2] </math> solving
<center>
<math>
1=k \int d\epsilon \, p(\epsilon) \left(\frac{t}{\epsilon}\right)^{2 \beta} \equiv k I(\beta).
</math>
</center> </li>
</ul>
</li><br>
<li><em> The critical disorder. </em> Consider now local fields <math> \epsilon </math> taken from a uniform distribution in <math> [-W/2, W/2] </math>.  Compute <math> I(\beta) </math> and show that it is non monotonic, with a local minimum <math> \beta^* </math> in the interval <math> [0, 1/2]</math>. Show that  <math> I(\beta^*) </math> increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when  <math> I(\beta^*)=k^{-1} </math>. Show that this gives
<center>
<math>
W_c = t \, 2 k e \log \left( \frac{W_c}{2 t}\right) \sim  t \, 2 k e \log \left(k\right)
</math>
</center>
Why the critical disorder increases with <math> k </math>?
  </li>
</ol>
<br>




Revision as of 19:27, 3 March 2024



Goal: in this final set of problems, we discuss the interplay between localization and glassiness, by connecting the solution to the Anderson problem on the Bethe lattice with the statistical physics problem of a directed polymer in random media on trees.
Techniques:


Problem 7.2: localization-delocalization transition on the Bethe lattice

We now focus on the self energies, since the criterion for localization is given in terms of these quantities. In this Problem we will determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs.


  1. The “localized" solution. We set and . Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:

    Justify why the solution corresponding to localization, , is always a solution when ; moreover, in the localized phase when is finite but small one has . How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?


  2. Imaginary approximation and distributional equation. We consider the equations for and neglect the terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_b } in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a \sim \eta \ll 1 } . Finally, we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ab} \equiv t } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=0 } for simplicity. Show that under these assumptions the probability density for the imaginary part, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma)} , satisfies

    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma)= \int \prod_{b=1}^k d\epsilon_b\,p(\epsilon_b)\int \prod_{c=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - t^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ \epsilon_b^2} \right) }

    Show that the Laplace transform of this distribution, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) } , satisfies

    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s)= \left[ \int d\epsilon\, p(\epsilon) e^{-\frac{s t^2 \eta}{\epsilon^2}} \Phi \left(\frac{s t^2 }{\epsilon^2} \right) \right]^k }


  3. The stability analysis. We now wish to check the stability of our assumption to be in the localized phase, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a \sim \eta \ll 1 } , which led to the identity above for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s) } . Our assumption is that the typical value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a } is small, except for cases in which one of the descendants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } is such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_b } is very small, in which case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a \sim 1/ \epsilon_b^2 } .
    • Show that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma \sim 1/ \epsilon^2 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(\epsilon)} is not gapped around zero, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma) \sim \Gamma^{-3/2}} , i.e. the distribution has tails contributed by these events in which the local fields happen to be very small.
    • Assume more generally that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}} for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in [1, 3/2]} . Show that this corresponds to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s) \sim 1- A |s|^\beta } for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } small, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta= \alpha-1 \in [0, 1/2] } .
    • Show that the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s) } gives for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1- A s^\beta =1- A k \int d\epsilon \, p(\epsilon) \frac{s^\beta t^{2 \beta}}{\epsilon^{2 \beta}}+ o(s^\beta) } , and therefore this is consistent provided that there exists a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta \in [0, 1/2] } solving

      Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=k \int d\epsilon \, p(\epsilon) \left(\frac{t}{\epsilon}\right)^{2 \beta} \equiv k I(\beta). }


  4. The critical disorder. Consider now local fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } taken from a uniform distribution in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-W/2, W/2] } . Compute and show that it is non monotonic, with a local minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^* } in the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, 1/2]} . Show that increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\beta^*)=k^{-1} } . Show that this gives

    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_c = t \, 2 k e \log \left( \frac{W_c}{2 t}\right) \sim t \, 2 k e \log \left(k\right) }

    Why the critical disorder increases with ?



the directed polymer treatment: KPP (es 1) es 2: The connection to directed polymer: linearisation and stability. Glassiness vs localization -->

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