L-4

Goal : final lecture on KPZ and directed polymers at finite dimension. We will show that for $d>2$ a "glass transition" takes place.

Directed Polymer in finite dimension

State of the Art

The directed polymer in random media belongs to the KPZ universality class. The behavior of this system is well understood in one dimension and in the mean-field case, more precisely for the directed polymer on the Cayley tree. In particular:

• In $d=1$, we have $\theta =1/3$ and a glassy regime present at all temperatures. The model is integrable through a non-standard Bethe Ansatz, and the distribution of $E_{\min }$ for a given boundary condition is of the Tracy–Widom type.

• In $d=\infty$, for the Cayley tree, an exact solution exists, predicting a freezing transition to a 1RSB phase ($\theta =0$).

In finite dimensions greater than one, no exact solutions are available. Numerical simulations indicate $\theta >0$ in $d=2$ and a glassy regime present at all temperatures. The case $d>2$ remains particularly intriguing.

First Moment

${\overline {Z(x,t)}}=\int _{x(0)=0}^{x(t)=x}{\mathcal {D}}x(\tau )\exp {\Big [}-{\frac {1}{T}}\int _{0}^{t}d\tau {\frac {1}{2}}(\partial _{\tau }x)^{2}{\Big ]}{\overline {\exp {\Big [}-{\frac {1}{T}}\int _{0}^{t}d\tau V(x(\tau ),\tau ){\Big ]}}}$

Due to the short-distance divergence of $\delta ^{d}(0)$,

$T^{2}{\overline {W^{2}}}=\int d\tau _{1}d\tau _{2}{\overline {V(x,\tau _{1})V(x,\tau _{2})}}=Dt\delta _{0}.$

Hence,

${\overline {Z(x,t)}}={\frac {1}{(2\pi tT)^{d/2}}}\exp {\Big [}-{\frac {x^{2}}{2tT}}{\Big ]}\exp {\Big [}{\frac {Dt\delta _{0}}{2T^{2}}}{\Big ]}=Z_{free}(x,t,T)\exp {\Big [}{\frac {Dt\delta _{0}}{2T^{2}}}{\Big ]}.$

Second Moment

For the second moment we need two replicas:

Step 1

${\overline {Z(x,t)^{2}}}=\int {\mathcal {D}}x_{1}\int {\mathcal {D}}x_{2}\exp \!{\Bigg [}-{\frac {1}{2T}}\int _{0}^{t}d\tau {\Big (}(\partial _{\tau }x_{1})^{2}+(\partial _{\tau }x_{2})^{2}{\Big )}{\Bigg ]}\;{\overline {\exp \!{\Bigg [}-{\frac {1}{T}}\int _{0}^{t}d\tau V(x_{1}(\tau ),\tau )-{\frac {1}{T}}\int _{0}^{t}d\tau V(x_{2}(\tau ),\tau ){\Bigg ]}}}.$

Step 2: Wick’s Theorem

${\overline {Z(x,t)^{2}}}=\exp \!{\Bigg [}{\frac {Dt\delta _{0}}{T^{2}}}{\Bigg ]}\int {\mathcal {D}}x_{1}\int {\mathcal {D}}x_{2}\exp \!{\Bigg [}-{\frac {1}{2T}}\int _{0}^{t}d\tau {\Big (}(\partial _{\tau }x_{1})^{2}+(\partial _{\tau }x_{2})^{2}-{\frac {D}{T^{2}}}\delta ^{d}[x_{1}(\tau )-x_{2}(\tau )]{\Big )}{\Bigg ]}.$

Step 3: Change of Coordinates

Let $X=(x_{1}+x_{2})/2$ and $u=x_{1}-x_{2}$. Then:

${\overline {Z(x,t)^{2}}}=({\overline {Z(x,t)}})^{2}{\frac {\displaystyle \int _{u(0)=0}^{u(t)=0}{\mathcal {D}}u\exp \!{\Bigg [}-\int _{0}^{t}d\tau {\frac {1}{4T}}(\partial _{\tau }u)^{2}-{\frac {D}{T^{2}}}\delta ^{d}[u(\tau )]{\Bigg ]}}{Z_{free}(u=0,t,2T)}}.$

Here,

$Z_{free}^{2}(x,t,T)=Z_{free}(X=x,t,T/2)\,Z_{free}(u=0,t,2T),\qquad Z_{free}(u=0,t,2T)=(4\pi Tt)^{d/2}.$

Two-Replica Propagator

Define the propagator:

$W(0,t)=\int _{u(0)=0}^{u(t)=0}{\mathcal {D}}u\exp {\Big [}-\int _{0}^{t}d\tau {\frac {1}{4T}}(\partial _{\tau }u)^{2}-{\frac {D}{T^{2}}}\delta ^{d}[u(\tau )]{\Big ]}.$

By the Feynman-Kac formula:

$\partial _{t}W(x,t)=-{\hat {H}}W(x,t),\quad {\hat {H}}=-T\nabla ^{2}-{\frac {D}{T^{2}}}\delta ^{d}[u].$

For $d>2$, The low-energy behavior depends on $D/T^{2}$:

• High temperature: the spectrum is positive and continuous, annealed and quenched coincide.

• Low temperature: bound states appear,the quenched free energy is smaller than the annealed one.

Let's do replica!

To make progress in disordered systems, we need to analyze the moments of the partition function. The first moment provide the annealed average and the second moment tell us about the fluctuantions. In particular, the partition function is self-averaging if

${\frac {\overline {Z(x,t)^{2}}}{({\overline {Z(x,t)}})^{2}}}=1\,.$

In this case annealed and the quenched average coincides in the thermodynamic limit. This strict condition is sufficient, but not necessary. What is necessary is to show that for large t

${\frac {\overline {Z(x,t)^{2}}}{({\overline {Z(x,t)}})^{2}}}<{\text{const}}$,

In the following, we compute these moments via a replica calculation, considering polymers starting at $0$ and ending at $x$.

To proceed, we only need two ingredients:

• The random potential $V(x,\tau )$ is a Gaussian field characterized by

${\overline {V(x,\tau )}}=0,\qquad {\overline {V(x,\tau )V(x',\tau ')}}=D\,\delta ^{d}(x-x')\,\delta (\tau -\tau ').$

• Since the disorder is Gaussian, averages of exponentials can be computed using Wick’s theorem:

${\overline {\exp(W)}}=\exp \!{\Big [}{\overline {W}}+{\frac {1}{2}}{\big (}{\overline {W^{2}}}-{\overline {W}}^{2}{\big )}{\Big ]},$

for any Gaussian random variable $W$.

These two properties are all we need to carry out the replica calculation below.