L-3

Goal: 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

We consider the following potential energy

$E_{pot}=\int dr{\frac {\sigma }{2}}(\nabla h)^{2}+V(r,h)$

The first term represents the elasticity of the manifold and the second term is the quenched disorder, due to the impurities. In general, the medium is D-dimensional, the internal coordinate of the manifold is d-dimensional and the height filed is N-dimensional. Hence,the following equations always holds:

$D=d+N$

In practice, we will study two cases:

• Directed Polymers ($d=1$), $D=1+N$. Examples are vortices, fronts...
• Elastic interfaces ($N=1$), $D=d+1$. Examples are domain walls...

Today we restrict to polymers. Note that they are directed because their configuration $h(r)$ is uni-valuated.

Directed polymers and Quantum Mechanics

It is useful to re-write the model using the following change of variable

$h\to x,\quad r\to t$

To fix the idea we can consider polymers of length $t$, starting in $x_{0}$ and ending in $x_{t}$. We sum over all possible polymers to compute the partition function at temperature $T=1$

$Z[x_{t},t;x_{0},0]=\int _{x(0)=x_{0}}^{x(t)=x_{t}}{\cal {D}}x\exp \left[-\int _{0}^{t}d\tau (\partial _{\tau }x)^{2}+V(x,\tau )\right]$

The previous equation gives the path integral expression of the propagator quantum particle, evolving in imaginary time. The Hamiltonian of the particle is

${\hat {H}}=-{\frac {d^{2}}{dx^{2}}}+V(x,\tau )$

And the partition function is the solution of the Schrodinger-like equation:

$\partial _{t}Z=-{\hat {H}}Z={\frac {d^{2}Z}{dx^{2}}}-V(x,\tau )Z,\quad {\text{with}}\;Z[x_{t},t=0;x_{0},0]=\delta (x-x_{0})$

DISCUTERE CON SATYA LEGAMI FEYMAN KAC FORMULA In this equation the noise in multiplicative and not additive as in the previous lecture. However some of the work we did last lecture will be useful today, thank to a miracoul transformation the Cole Hopf transformation.