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Non-crossing polymers and the KPZ equation Andrea De Luca in - - PowerPoint PPT Presentation
Non-crossing polymers and the KPZ equation Andrea De Luca in - - PowerPoint PPT Presentation
01/07/2015 1 29 / Non-crossing polymers and the KPZ equation Andrea De Luca in collaboration with P. Le Doussal arXiv:1505.04802 01/07/2015 KPZ equation PRL 56 889 (1986), Kardar, Parisi, Zhang relaxation lowest order Gaussian noise
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Concrete examples 3 29 / 01/07/2015
Turbulent liquid crystals - PRL 104 230601
Tracy-Widom distribution
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Cole-Hopf mapping 4 29 / 01/07/2015
diffusion equation in a random potential: directed polymer partition function
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Quantum mechanics and replica 5 29 / 01/07/2015
Path integral representation (Feynman - Kac)
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High-temperature and Lieb-Liniger 6 29 / 01/07/2015
Rescaling of variables We end up with the attractive Lieb-Liniger Hamiltonian which is integrable in 1dimension!
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Bethe-ansatz approach 7 29 / 01/07/2015
Hard to treat: it contains space-time correlation of the KPZ height
decomposition in eigenstates
If we can compute the spectrum, we can find arbitrary moments...
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Bethe-ansatz equations 8 29 / 01/07/2015
The initial condition is symmetric: the dynamics lies in the bosonic sector of the Hamiltonian The coefficient implements the scattering matrix How to fix the values of rapidities?
superposition of plane waves in each sector
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Periodic boundary condition 9 29 / 01/07/2015
The values of rapidities are fixed by boundary
- conditions. In the symplest case
Bethe-Ansatz equations for the LL model Solutions at finite L are not easy... But in the thermodynamic limit?
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String ansatz 10 29 / 01/07/2015
If for large L, we have a divergence in the LHS, which must be compensated by a pole in the RHS
bound states (strings) 5-string
1-string
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String features 11 29 / 01/07/2015
5-string energy momentum
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Needed ingredients 12 29 / 01/07/2015
Norm WF
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General expression for moments 13 29 / 01/07/2015
The sum over eigenstates becomes the sum
- ver the possible partitioning of the n particles
into strings
sum over partitions
It is exact... but how to deal with it?
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Partition function at fixed string number 14 29 / 01/07/2015
Use the grancanonical partition function: In this way we can recover the free energy distribution
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Fredholm determinant 15 29 / 01/07/2015
Exchanging the two sums, we obtain In the large time limit, one obtains
Tracy-Widom GUE distribution EPL 90 2 (2010) Calabrese, Le Doussal, Rosso
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Non-crossing polymers 16 29 / 01/07/2015
Can we use replica approach to treat non-crossing polymers? Simplest example of interaction, together with disorder...!
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Karlin-McGregor formula 17 29 / 01/07/2015
Similar formulas for more than two polymers
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Coinciding points 18 29 / 01/07/2015
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Replica for non-crossing polymers 19 29 / 01/07/2015
The expression is analogous to the one for single polymer. But the bosonic sector gives a vanishing contribution! How to build wave functions with different symmetries?
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Wave function and Young tableau 20 29 / 01/07/2015
We look for eigen functions antisymmetric in the first two variables...
symmetric antisymmetric
More general ansatz...
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Nested Bethe Ansatz 21 29 / 01/07/2015
The auxliary variable implement the symmetry
- f the wave function.
In general, one auxiliary variable for every doubled column
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String ansatz? 22 29 / 01/07/2015
For large L, the first equation suggests again the presence of strings What about the auxiliary variable?
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Contour integral 23 29 / 01/07/2015
The solution of the second equation are non trivial... But we are only interested on the sum
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Comparison with bosonic case 24 29 / 01/07/2015
After summing over the auxiliary variable, we get an expression very similar to the bosonic case Norm is unchanged
conserved quantities
- f the LL
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Generalized Gibbs Ensemble 25 29 / 01/07/2015
we replace time evolution with a generalized evolution with multiples times
The average non-crossing probability is not affected by disorder!
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Comparison with numerics 26 29 / 01/07/2015
Two-lines derivation
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Recipe for higher moments 27 29 / 01/07/2015
In order to compute higher moments
- symmetrize the polynomial in terms of
conserved charges
- write the result as a set of derivatives
applied to the generalized moments
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Results for higher moments 28 29 / 01/07/2015
Leading order
Physical picture:
- for most of the realization:
p is exponentially small
- for a fraction 1/t
- f the realization, p is O(c^2)
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Conclusions 29 29 / 01/07/2015
- We developed a framework based on the Nested Bethe ansatz
to deal with non-crossing polymers in random media;
- We computed exactly the large times asymptotics
for the moments of the non-crossing probability for two polymers;
- Agreement with numerical lattice simulations:
the crossing probability is most of the time exponentially small Open questions:
- generalization to multi-polymers
- higher order large time asymptotics: connection with