TBan-II - Revision history

Rosso at 02:44, 16 September 2025

2025-09-16T02:44:16Z

Rosso: /* Directed Polymers on a lattice */

2025-08-30T18:31:04Z

Directed Polymers on a lattice

← Older revision		Revision as of 20:31, 30 August 2025
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	</math></center>		</math></center>

	<strong>Universal exponents: </strong> Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions. ~~Note that <math> \omega= \theta </math>, while for an interface <math> \omega= d \theta </math>.~~		<strong>Universal exponents: </strong> Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions.

	<strong>Non-universal constants: </strong> <math> c_\infty,b_\infty, a_\infty </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!		<strong>Non-universal constants: </strong> <math> c_\infty,b_\infty, a_\infty </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!

Rosso: /* Dijkstra Algorithm and transfer matrix */

2025-08-30T18:29:38Z

Dijkstra Algorithm and transfer matrix

← Older revision		Revision as of 20:29, 30 August 2025
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	=~~=Dijkstra Algorithm and transfer matrix=~~=		=Directed Polymers on a lattice=


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	We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementation of the powerful Dijkstra algorithm.		We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementation of the powerful Dijkstra algorithm. Dijkstra allows to identify the minimal energy among the exponential number of configurations <math> x(\tau)</math>

	Dijkstra allows to identify the minimal energy among the exponential number of configurations <math> x(\tau)</math>
	<center> <math>		<center> <math>
	E_{\min} = \min_{x(\tau)} E[x(\tau)].		E_{\min} = \min_{x(\tau)} E[x(\tau)].

Rosso: /* Entropy and scaling relation */

2025-08-30T18:28:30Z

Entropy and scaling relation

← Older revision		Revision as of 20:28, 30 August 2025
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	\theta=2 \zeta-1		\theta=2 \zeta-1
	</math></center>		</math></center>
	~~We will see that this~~ relation is actually exact.		This relation is actually exact also for the continuum model.

Rosso: Created page with "==Dijkstra Algorithm and transfer matrix== [[File:SketchDPRM.png|thumb|left|Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left either one step right. A random energy $V(\tau,x)$ is associated at each node and the total energy is simply $E[x(\tau)] =\sum_{\tau=0}^t V(\tau,x)$ . ]] We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementa..."

2025-08-30T18:26:39Z

Created page with "==Dijkstra Algorithm and transfer matrix== thumb|left|Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left either one step right. A random energy <math> V(\tau,x)</math> is associated at each node and the total energy is simply <math> E[x(\tau)] =\sum_{\tau=0}^t V(\tau,x)</math>. We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementa..."

New page

==Dijkstra Algorithm and transfer matrix==

[[File:SketchDPRM.png|thumb|left|Sketch of the discrete Directed Polymer model. At each time the polymer grows either one step left either one step right. A random energy <math> V(\tau,x)</math> is associated at each node and the total energy is simply <math> E[x(\tau)] =\sum_{\tau=0}^t V(\tau,x)</math>. ]]

We introduce a lattice model for the directed polymer (see figure). In a companion notebook we provide the implementation of the powerful Dijkstra algorithm.

Dijkstra allows to identify the minimal energy among the exponential number of configurations <math> x(\tau)</math>
<center> <math>
E_{\min} = \min_{x(\tau)} E[x(\tau)].
</math></center>

We are also interested in the ground state configuration <math> x_{\min}(\tau) </math>.
For both quantities we expect scale invariance with two exponents <math> \theta, \zeta</math> for the energy and for the roughness
<center>
<math>
E_{\min} = c_\infty t + b_\infty t^{\theta}\chi, \quad x_{\min}(t/2)) \sim a_\infty t^{\zeta} \tilde \chi
</math></center>

Universal exponents: Both <math> \theta, \zeta </math> are Independent of the lattice, the disorder distribution, the elastic constants, or the boudanry conditions. Note that <math> \omega= \theta </math>, while for an interface <math> \omega= d \theta </math>.

Non-universal constants: <math> c_\infty,b_\infty, a_\infty </math> are of order 1 and depend on the lattice, the disorder distribution, the elastic constants... However <math> c_\infty </math> is independent on the boudanry conditions!

Universal distributions: <math> \chi, \tilde \chi </math> are instead universal, but depends on the boundary condtions. Starting from 2000 a magic connection has been revealed between this model and the smallest eigenvalues of random matrices. In particular I discuss two different boundary conditions:

* Droplet: <math> x(\tau=0) = x(\tau=t) = 0 </math>. In this case, up to rescaling, <math> \chi</math> is distributed as the smallest eigenvalue of a GUE random matrix (Tracy Widom distribution <math>F_2(\chi) </math>)

* Flat: <math> x(\tau=0) = 0 </math> while the other end <math> x(\tau=t) </math> is free. In this case, up to rescaling, <math> \chi</math> is distributed as the smallest eigenvalue of a GOE random matrix (Tracy Widom distribution <math>F_1(\chi) </math>)
===Entropy and scaling relation===

It is useful to compute the entropy
<center>
<math>
\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{t} +O(x^4)
</math></center>
From which one could guess from dimensional analysis
<center>
<math>
\theta=2 \zeta-1
</math></center>
We will see that this relation is actually exact.