L-4

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.

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.

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 $\theta >0$.

Back to REM: condensation of the Gibbs measure

Thanks to the computation of ${\overline {n(x)}}$, we can identify an important fingerprint of the glassy phase. Let's compare the weight of the ground state against the weight of all other states:

${\frac {\sum _{\alpha }z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}{\frac {z_{\alpha }}{z_{\alpha _{\min }}}}=1+\sum _{\alpha \neq \alpha _{\min }}e^{-\beta (E_{\alpha }-E_{\min })}\sim 1+\int _{0}^{\infty }dx\,{\frac {d{\overline {n(x)}}}{dx}}\,e^{-\beta x}=1+\int _{0}^{\infty }dx\,{\frac {e^{x/b_{M}}}{b_{M}}}\,e^{-\beta x}$

Behavior in Different Phases:

• High-Temperature Phase ($\beta <\beta _{c}=1/b_{M}={\sqrt {2\log 2}}$):

In this regime, the total weight of the excited states dominates over the weight of the ground state. The ground state is therefore not deep enough to overcome the finite entropy contribution. As a result, the probability of sampling the same configuration twice from the Gibbs measure is exponentially small in the system size.

• Low-Temperature Phase ($\beta >\beta _{c}=1/b_{M}={\sqrt {2\log 2}}$):

In this regime, the integral is finite:

$\int _{0}^{\infty }dx\,e^{(1/b_{M}-\beta )x}/b_{M}={\frac {1}{\beta b_{M}-1}}={\frac {\beta _{c}}{\beta -\beta _{c}}}$

In this regime, the total weight of the excited states is of the same order as the weight of the ground state. Consequently, the ground state is occupied with finite probability, reminiscent of Bose–Einstein condensation. However, unlike the directed polymer in finite dimension, this condensation involves not only the ground state but also the first excited states.

Overlap Distribution and Replica Symmetry Breaking:

The structure of states can be further characterized through the overlap between two configurations $\alpha$ and $\gamma$, defined as

$q_{\alpha ,\gamma }={\frac {1}{N}}\sum _{i=1}^{N}\sigma _{i}^{\alpha }\sigma _{i}^{\gamma },$

which takes values in the interval $(-1,1)$. The distribution $P(q)$ of the overlap between two configurations sampled from the Gibbs measure distinguishes the two phases:

At high temperature ($\beta <\beta _{c}$), the system is replica symmetric and the overlap distribution is concentrated at zero:

$P(q)=\delta (q).$

At low temperature ($\beta >\beta _{c}$), the system exhibits one-step replica symmetry breaking, and the overlap distribution becomes

$P(q)={\tfrac {\beta _{c}}{\beta }}\,\delta (q)+{\Bigl (}1-{\tfrac {\beta _{c}}{\beta }}{\Bigr )}\,\delta (1-q).$

Finite Dimensional Systems

In finite dimensions, the fluctuations of the ground-state energy are characterized by an exponent $\theta$:

${\overline {{\big (}E_{\min }-{\overline {E_{\min }}}{\big )}^{2}}}\sim L^{2\theta },$

where $L$ is the linear size of the system and $N=L^{D}$ is the number of degrees of freedom.

When $\theta <0$, the critical temperature vanishes with increasing system size, leading to the absence of a glass transition. This scenario occurs in many low-dimensional systems, such as the Edwards–Anderson model in two dimensions.

When $\theta >0$, one must extend the definition of this exponent to finite temperatures and consider the fluctuations of the free energy $F(L,\beta )$. Three representative cases are:

• Directed polymer in $N=1,2$:

The fluctuations of the ground state exhibit a positive, temperature-independent exponent $\theta$. In this situation, only the glassy phase exists, and

$P(q)=\delta (1-q),$

because producing an excitation with vanishing overlap with the ground state is very costly.

• Directed polymer in $N=3$:

The exponent $\theta$ depends on the temperature: it vanishes above the glass transition and becomes strictly positive below it. Accordingly,

$P(q)=\delta (1-q)$ at low temperature, and $P(q)=\delta (q)$ at high temperature.

• Directed polymer on the Cayley tree:

The behavior is analogous to the Random Energy Model: $\theta =0$ in both phases. At high temperature,

$P(q)=\delta (q),$

while at low temperature the system exhibits the one-step replica symmetry breaking picture:

$P(q)={\tfrac {\beta _{c}}{\beta }}\,\delta (q)+{\Bigl (}1-{\tfrac {\beta _{c}}{\beta }}{\Bigr )}\,\delta (1-q).$