L-3 - Revision history

Rosso at 20:02, 1 March 2026

2026-03-01T20:02:52Z

Rosso: /* Cole Hopf Transformation */

2026-01-31T09:58:16Z

Cole Hopf Transformation

← Older revision		Revision as of 11:58, 31 January 2026
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	At low temperature, the free energy approaches the ground state energy, <math>E_{\min}</math>.		At low temperature, the free energy approaches the ground state energy, <math>E_{\min}</math>.

	~~{\| class="wikitable"~~
	~~\|+ Dictionary~~
	\|-
	~~! KPZ !! KPZ exponents !! Directed polymer !! Directed polymer exponents~~
	\|-
	~~\| <math> r </math>\|\| <math> r \sim t^{1/z}</math>\|\| <math>x </math>\|\| <math>x\sim t^{\zeta}</math>~~
	\|-
	~~\| <math>t</math> \|\|<math>h(r,t) \sim t^{\alpha/z}</math> \|\| <math>t</math>\|\|~~
	\|-
	~~\| <math>h</math> \|\| <math>h(r,t) \sim r^{\alpha}</math>\|\| <math>F, E_{\min}</math> \|\| <math> \overline{(E_{\min} - \overline{E_{\min}})^2} \sim t^{2\theta} </math>~~
	\|}

	{\| class="wikitable"		{\| class="wikitable"

Rosso: /* Cole Hopf Transformation */

2026-01-31T09:57:54Z

Cole Hopf Transformation

← Older revision		Revision as of 11:57, 31 January 2026
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	\| <math>h(r,t) \sim t^{\alpha/z}</math>		\| <math>h(r,t) \sim t^{\alpha/z}</math>
	\| <math>t</math>		\| <math>t</math>
	\| <math>F,\,E_{\min}</math>		\| <math>\overline{(E_{\min}-\overline{E_{\min}})^2} \sim t^{2\theta}</math>
	\|-		\|-
	\| <math>h</math>		\| <math>h</math>

Rosso: /* Cole Hopf Transformation */

2026-01-31T09:55:47Z

Cole Hopf Transformation

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 | <math>h</math> || <math>h(r,t) \sim r^{\alpha}</math>|| <math>F, E_{\min}</math> || <math> \overline{(E_{\min} - \overline{E_{\min}})^2} \sim t^{2\theta} </math>
 |}
 We conclude that

Rosso: /* Exercise: free particle in 1D */

2026-01-31T09:52:13Z

Exercise: free particle in 1D

← Older revision		Revision as of 11:52, 31 January 2026
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	</center>		</center>


	~~====Exercise: free particle in 1D====~~
	* <Strong> Step 1: The spectrum</Strong>
	~~For a free particle there a no discrete eigenvalue, but only a continuum spectrum. In 1D the hamiltonian is <math> \hat H =-(T/2) \partial_x^2</math>.~~
	~~Show that the continuum spectrum has the form~~
	~~<center><math> \psi_k(x) =\frac{1}{\sqrt{2 \pi}} e^{i k x} ,\quad \text{with} \; E_k =\frac{ T k^2}{2} </math></center>~~
	Here <math>k </math> is a real number, while <math>E_k</math> is a positive number. Note that the states are delocalized on the entire real axis and they cannot be normalized to unit but you impose the Dirac delta normalization:
	~~<center><math> \int_{-\infty}^\infty \, dx \, \psi_{k'}^*(x)\psi_k(x) = \delta(k-k') </math></center>~~
	* <Strong> Step 2: the DOS </Strong>
	~~For a given energy <math> E </math> there are two states~~
	~~<center><math> \psi_E^{-}(x) =\frac{1}{\sqrt{2 \pi}} e^{-i \sqrt{2 E/T} x} ,\quad \text{and} ,\quad \psi_E^{+}(x) =\frac{1}{\sqrt{2 \pi}} e^{i \sqrt{2 E/T} x} </math></center>~~
	~~Use the definition of DOS <center><math> \rho(E)= \int_{-\infty}^\infty \, dk \,\delta(E-E_k) </math></center> ans show that for both <math> \psi_E^{-}(x) , \psi_E^{-}(x) </math> you have :~~
	~~<center><math> \rho(E) = \frac{1}{\sqrt{2 E T} } </math></center>~~
	* <Strong> Step 3: the propagator </Strong>
	~~Use the spectral decomposition of the propagator to recover <math> Z_{\text{free}}(x,t)</math>. <Strong>~~
	~~Tip:</Strong> use <math> \int_{-\infty}^\infty \, dx \, \exp[-(a x^2 +b x)]= \sqrt{\pi/a} \exp[b^2/(4 a)] </math>.~~

	== Cole Hopf Transformation==		== Cole Hopf Transformation==

Rosso: /* Remarks */

2026-01-31T09:51:39Z

Remarks

← Older revision		Revision as of 11:51, 31 January 2026
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	</math></center>		</math></center>

			====Hints: free particle in 1D====

			For a free particle in one dimension the Hamiltonian is
			<math>\hat H = -\frac{T}{2}\,\partial_x^2</math>.

			'''Spectrum.'''
			The spectrum is purely continuous. The eigenstates are plane waves
			<center>
			<math>
			\psi_k(x)=\frac{1}{\sqrt{2\pi}}e^{ikx},
			\qquad
			E_k=\frac{T k^2}{2},
			</math>
			</center>
			with <math>k\in\mathbb{R}</math>. The states are delocalized and satisfy Dirac delta normalization
			<center>
			<math>
			\int_{-\infty}^{\infty} dx\,\psi_{k'}^*(x)\psi_k(x)=\delta(k-k').
			</math>
			</center>

			'''Energy representation and density of states.'''
			For a given energy <math>E>0</math> there are two degenerate states,
			<center>
			<math>
			\psi_E^{\pm}(x)=\frac{1}{\sqrt{2\pi}}\,e^{\pm i\sqrt{2E/T}\,x}.
			</math>
			</center>
			The density of states is obtained from
			<center>
			<math>
			\rho(E)=\int_{-\infty}^{\infty} dk\,\delta(E-E_k),
			\qquad
			E_k=\frac{T k^2}{2}.
			</math>
			</center>

			'''Propagator.'''
			Using the spectral decomposition one can write
			<center>
			<math>
			Z(x,t)
			=\int_0^{\infty} dE\,\rho(E)
			\sum_{\sigma=\pm}
			\psi_E^{\sigma}(0)\psi_E^{\sigma *}(x)\,e^{-Et}.
			</math>
			</center>
			Evaluating the resulting Gaussian integral yields
			<center>
			<math>
			Z_{\text{free}}(x,t)=\frac{e^{-x^2/(2Tt)}}{\sqrt{2\pi Tt}}.
			</math>
			</center>

			Useful identity:
			<center>
			<math>
			\int_{-\infty}^{\infty} dx\,e^{-(a x^2+b x)}
			=\sqrt{\frac{\pi}{a}}\,e^{\,b^2/(4a)},\qquad a>0.
			</math>
			</center>

Rosso: /* Feynman-Kac formula */

2026-01-31T09:46:55Z

Feynman-Kac formula

← Older revision		Revision as of 11:46, 31 January 2026
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	</math></center>		</math></center>

	~~== Feynman-Kac formula==~~
	~~Let's derive the Feyman Kac formula for <math>Z(x,t)</math> in the general case:~~
	* First, focus on free paths and introduce the following probability
	~~<center> <math>~~
	~~P[A,x,t] =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) \exp\left[- \frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 \right] \delta\left( \int_0^t d \tau V(x(\tau),\tau)-A \right)~~
	~~</math></center>~~
	* Second, the moments generating function
	~~<center> <math>~~
	~~Z_p(x ,t) = \int_{-\infty}^\infty d A e^{-p A} P[A,x,t] =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) e^{-\frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 -p \int_0^t d \tau V(x(\tau),\tau)}~~
	~~</math></center>~~
	* Third, the backward approach. Consider free paths evolving up to <math>t+dt</math> and reaching <math>x</math> :
	~~<center> <math>~~
	Z_p(x,t+dt)= \left\langle e^{-p \int_0^{t+dt} d \tau V(x(\tau),\tau)}\right\rangle= \left\langle e^{-p \int_0^{t} d \tau V(x(\tau),\tau)}\right\rangle e^{-p V(x,t) d t } =[1 -p V(x,t) d t +\dots]\left\langle Z_p(x-\Delta x,t) \right\rangle_{\Delta x}
	~~</math></center>~~
	Here <math> \langle \ldots \rangle</math> is the average over all free paths, while <math> \langle \ldots \rangle_{\Delta x}</math> is the average over the last jump, namely <math> \langle \Delta x \rangle=0
	~~</math> and <math> \langle \Delta x^2 \rangle=T d t </math>.~~
	* At the lowest order we have
	~~<center> <math>~~
	~~Z_p(x,t+dt)= Z_p(x,t) +dt \left[ \frac{T}{2} \partial_x^2 Z_p -p V(x,t) Z_p \right] +O(dt^2)~~
	~~</math></center>~~
	~~Replacing <math> p=1/T</math> we obtain the partition function is the solution of the Schrodinger-like equation:~~
	~~<center> <math>~~
	~~\partial_t Z(x,t) =- \hat H Z = - \left[ -\frac{T}{2}\frac{d^2 }{d x^2} + \frac{V(x,\tau)}{T}\right] Z(x,t)~~
	~~</math>~~
	~~<math>~~
	~~Z[x,t=0]=\delta(x)~~
	~~</math>~~
	~~</center>~~

	~~===Remarks===~~
	~~<Strong>Remark 1:</Strong>~~

	This equation is a diffusive equation with multiplicative noise. Edwards Wilkinson is instead a diffusive equation with additive noise. The Cole Hopf transformation allows to map the diffusive equation with multiplicative noise in a non-linear equation with additive noise. We will apply this tranformation and have a surprise.

	~~<Strong>Remark 2:</Strong>~~
	This hamiltonian is time dependent because of the multiplicative noise <math>V(x,\tau)/T</math>. For a <Strong> time independent </Strong> hamiltonian, we can use the spectrum of the operator. In general we will have to parts:
	* A discrete set of eigenvalues <math>E_n</math> with the eigenstates <math>\psi_n(x)</math>
	* A continuum part where the states <math>\psi_E(x)</math> have energy <math>E</math>. We define the density of states (DOS) <math>\rho(E)</math>, such that the number of states with energy in <math>(E, E+dE)</math> is <math>\rho(E) dE </math>.
	~~In this case <math> Z[x,t] </math> can be written has the sum of two contributions:~~
	~~<center> <math>~~
	~~Z[x,t] = \left( e^{- \hat H t} \right)_{0 \to x}= \sum_n \psi_n(0) \psi_n^(x) e^{- E_n t} + \int_0^\infty dE \, \rho(E) \psi_E(0) \psi_E^(x) e^{- E t}.~~
	~~</math>~~
	~~</center>~~

	====Exercise: free particle in 1D====		====Exercise: free particle in 1D====

Rosso: /* Back to the continuum model */

2026-01-31T09:45:31Z

Back to the continuum model

@@ Line 53: / Line 53: @@
 This relation is actually exact also for the continuum model.
-==Back to the continuum model==
+=Directed polymers in the continuum=
-Let us consider polymers <math>
-x(\tau) </math>  of length  <math>
+== Polymer partition function and propagator of a quantum particle==
-t </math>, starting in  <math>0 </math>  and ending in <math>x </math> and at thermal equlibrium at   temperature <math>T</math>. The partition function of the model writes as
 <center> <math>
 Z(x,t) =\int_{x(0)=0}^{x(t)=x} {\cal D} x(\tau) \exp\left[- \frac{1}{T} \int_0^t d \tau \frac{1}2(\partial_\tau x)^2 +V(x(\tau),\tau)\right]
 </math></center>
-For simplicity, we assume a white noise, <math> \overline{V(x,\tau) V(x',\tau')} = D \delta(x-x') \delta(\tau-\tau') </math>. Here, the partition function is written as a sum over all possible paths, corresponding to all possible polymer configurations that start at <math>0</math> and end at <math>x</math>, weighted by the appropriate Boltzmann factor.
+Here, the partition function is written as a sum over all possible paths, corresponding to all possible polymer configurations that start at <math>0</math> and end at <math>x</math>, weighted by the appropriate Boltzmann factor.
 Let's perform the following change of variables: <math>\tau=i t' </math>. We also identifies <math>T</math> with <math>\hbar</math> and <math> \tilde t= -i t </math> as the time.
@@ Line 73: / Line 91: @@
 From the Feymann path integral formulation, <math> Z[x,\tilde t]</math>  is the propagator of the quantum particle.
 In absence of disorder, one can find the propagator of the free particle, that, in the original variables, writes:
 <center> <math>

Rosso: /* Dijkstra Algorithm and transfer matrix */

2026-01-31T09:42:34Z

Dijkstra Algorithm and transfer matrix

← Older revision		Revision as of 11:42, 31 January 2026
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	Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.		Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.

	=~~=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>
			<center> <math>
	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>		E_{\min} = \min_{x(\tau)} E[x(\tau)].
			</math></center>

	We are also interested in the ground state configuration <math> x_{\min}(\tau) </math>.		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		For both quantities we expect scale invariance with two exponents <math> \theta, \zeta</math> for the energy and for the roughness
	<center>		<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>		<math>
			E_{\min} = c_\infty t + \kappa_1 t^{\theta}\chi, \quad x_{\min}(t/2)) \sim \kappa_2 t^{\zeta} \tilde \chi
			</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,\kappa_1, \kappa_2 </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>Universal distributions: </strong> <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:		<strong>Universal distributions: </strong> <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:
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	<center>		<center>
	<math>		<math>
	\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{t} +O(x^4)		\text{Entropy}= \ln\binom{t}{\frac{t-x}{2}} \approx t \ln 2 -\frac{x^2}{2 t} +O(x^4)
	</math></center>		</math></center>
	From which one could guess from dimensional analysis		From which one could guess from dimensional analysis
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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.

	==Back to the continuum model==		==Back to the continuum model==

Rosso at 09:40, 31 January 2026

2026-01-31T09:40:33Z

@@ Line 2: / Line 2: @@
 <strong>Goal: </strong> This lecture is dedicated to a classical model in disordered systems: the directed polymer in random media. It has been introduced to model vortices in superconductur or domain wall in magnetic film. We will focus here on the algorithms that identify the ground state or compute the free energy at temperature T, as well as, on the Cole-Hopf transformation that map this model on the KPZ equation.
-=Polymers, interfaces and manifolds in random media=
+Examples: vortex lines, DNA strands,  fronts.
-Today we restrict to polymers. Note that they are directed because their configuration <math>
+Although polymers may form loops, we restrict to directed polymers, i.e., configurations without overhangs or backward turns.
 ==Dijkstra Algorithm and transfer matrix==