Goal : final lecture on KPZ and directed polymers at finite dimension. We will show that for
a "glass transition" takes place.
KPZ : from 1d to the Cayley tree
We know a lot about KPZ, but there is still much to understand:
- In
, we have found
and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of quantities such as
. However, it does not determine the actual distribution of
for a given
. In particular, we have no clear understanding of the origin of the Tracy-Widom distribution.
- In
, an exact solution exists for the Cayley tree, predicting a freezing transition to a 1RSB phase (
).
- In finite dimensions greater than one, no exact solutions are available. Numerical simulations indicate
in
. The case
remains particularly intriguing.
Let's do replica!
To make progress in disordered systems, we need to analyze the moments of the partition function. From Valentina's lecture, recall that if
then the partition function is self-averaging, and
.
The condition above is sufficient but not necessary. It is enough that
,
when
, to ensure the equivalence between annealed and quenched averages.
In the following, we compute this quantity, which corresponds to a two-replica calculation. For simplicity, we consider polymers starting at
and ending at
. We recall that:
is a Gaussian field with
- From Wick's theorem, for a generic Gaussian field
, we have
The first moment
The first moment of the partition function is straightforward to compute and corresponds to a single replica:
Note that the term
exhibits a short-distance divergence due to the delta function. Hence, we can write:
The second moment
For the second moment, there are two replicas:
- Step 1: The second moment is
- Step 2: Using Wick's theorem, we obtain
and we can write:
- Step 3: Changing coordinates
, we get
where we used
with
The two replica propagator
Let us define the propagator:
Using the Feynman-Kac formula, we can write the following equation:
Here, the Hamiltonian is given by:
The Spectrum of the Two-Replica Hamiltonian
The single-particle potential is time-independent and attractive. Since it is time-independent, we can use the spectral decomposition of the propagator. The long-time behavior is controlled by the low-energy part of the spectrum. In the presence of an attractive potential, we may have:
- A discrete set of eigenvalues corresponding to bound states, followed by a continuous spectrum.
- Only a continuous spectrum.
As a funcion of the dimension we distiguish two cases:
- For
:
An attractive potential always leads to the formation of a bound state.The ground state has a negative energy
. At long times, the propagator behaves as:
This implies that at all temperatures, in the limit
:
- For
:
The low-energy part of the spectrum is controlled by the prefactor
. At high temperatures, the spectrum remains continuous and positive. At low temperatures, bound states appear. Thus, in the limit
:
This transition, in
, separates a high-temperature phase with
and a low-temperature phase with
and no RSB.