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 (surface tension) non-linearity In 1D, the renormalization group provides exact exponents 01/07/2015 2 29 /

Concrete examples Turbulent liquid crystals - PRL 104 230601 Tracy-Widom distribution 01/07/2015 3 29 /

Cole-Hopf mapping di ff usion equation in a random potential: directed polymer partition function 01/07/2015 4 29 /

Quantum mechanics and replica Path integral representation (Feynman - Kac) 01/07/2015 5 29 /

High-temperature and Lieb-Liniger Rescaling of variables We end up with the attractive Lieb-Liniger Hamiltonian which is integrable in 1dimension! 01/07/2015 6 29 /

Bethe-ansatz approach Hard to treat: it contains space-time correlation of the KPZ height decomposition in eigenstates If we can compute the spectrum, we can fi nd arbitrary moments... 01/07/2015 7 29 /

Bethe-ansatz equations The initial condition is symmetric: the dynamics lies in the bosonic sector of the Hamiltonian superposition of plane waves in each sector The coe ffi cient implements the scattering matrix How to fi x the values of rapidities? 01/07/2015 8 29 /

Periodic boundary condition The values of rapidities are fi xed by boundary conditions. In the symplest case Bethe-Ansatz equations for the LL model Solutions at fi nite L are not easy... But in the thermodynamic limit? 01/07/2015 9 29 /

String ansatz 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 01/07/2015 10 29 /

String features 5-string energy momentum 01/07/2015 11 29 /

Needed ingredients WF Norm 01/07/2015 12 29 /

General expression for moments The sum over eigenstates becomes the sum over the possible partitioning of the n particles into strings sum over partitions It is exact... but how to deal with it? 01/07/2015 13 29 /

Partition function at fi xed string number Use the grancanonical partition function: In this way we can recover the free energy distribution 01/07/2015 14 29 /

Fredholm determinant Exchanging the two sums, we obtain In the large time limit, one obtains Tracy-Widom EPL 90 2 (2010) GUE distribution Calabrese, Le Doussal, Rosso 01/07/2015 15 29 /

Non-crossing polymers Can we use replica approach to treat non-crossing polymers? Simplest example of interaction, together with disorder...! 01/07/2015 16 29 /

Karlin-McGregor formula Similar formulas for more than two polymers 01/07/2015 17 29 /

Coinciding points 01/07/2015 18 29 /

Replica for non-crossing polymers The expression is analogous to the one for single polymer. But the bosonic sector gives a vanishing contribution! How to build wave functions with di ff erent symmetries? 01/07/2015 19 29 /

Wave function and Young tableau We look for eigen functions antisymmetric in the fi rst two variables... symmetric antisymmetric More general ansatz... 01/07/2015 20 29 /

Nested Bethe Ansatz The auxliary variable implement the symmetry of the wave function. In general, one auxiliary variable for every doubled column 01/07/2015 21 29 /

String ansatz? For large L, the fi rst equation suggests again the presence of strings What about the auxiliary variable? 01/07/2015 22 29 /

Contour integral The solution of the second equation are non trivial... But we are only interested on the sum 01/07/2015 23 29 /

Comparison with bosonic case After summing over the auxiliary variable, we get an expression very similar to the bosonic case conserved quantities of the LL Norm is unchanged 01/07/2015 24 29 /

Generalized Gibbs Ensemble we replace time evolution with a generalized evolution with multiples times The average non-crossing probability is not a ff ected by disorder! 01/07/2015 25 29 /

Comparison with numerics Two-lines derivation 01/07/2015 26 29 /

Recipe for higher moments 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 01/07/2015 27 29 /

Results for higher moments Physical picture: Leading order - for most of the realization: p is exponentially small - for a fraction 1/t of the realization, p is O(c^2) 01/07/2015 28 29 /

Conclusions • 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 random matrices? 01/07/2015 29 29 /