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:

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

The single-particle potential is time-independent and attractive. Long-time behavior is governed by the low-energy eigenstates.

For $d\leq 2$, the attractive potential always produces a bound state with energy $E_{0}<0$. Hence, at long times:

$W(x,t)\sim e^{|E_{0}|t}$

This explosion means that the quenched free energy is smaller than the annealed one at all temperatures.

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

• High temperature: the spectrum is positive and continuous. Annealed and quenched coincide, the exponent $\theta =0$.

• Low temperature: bound states appear. No replica-symmetry breaking (RSB), but the quenched free energy is smaller than the annealed one. Numerical simulations show $\theta >0$.