Exact results from the replica Bethe ansatz from KPZ growth and random directed polymers
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1 Exact results from the replica Bethe ansatz from KPZ growth and random directed polymers P. Le Doussal (LPTENS) with : Pasquale Calabrese (Univ. Pise, SISSA) Alberto Rosso (LPTMS Orsay) Thomas Gueudre (LPTENS,Torino) most recent: Andrea de Luca (LPTENS,Orsay) Thimothee Thiery (LPTENS) - many discrete models in KPZ class exhibit universality related to random matrix theory: Tracy Widom distributions: of largest eigenvalue of GUE,GOE.. RBA: method integrable systems (Bethe Ansatz) +disordered systems(replica) - provides solution directly continuum KPZ eq./dp (at all times) - also to discrete models => allowed rigorous replica KPZ eq. is in KPZ class!
2 Outline: - KPZ equation, KPZ class, random matrices,tracy Widom distributions. - solving KPZ at any time by mapping to directed paths then using (imaginary time) quantum mechanics attractive bose gas (integrable) => large time TW distrib. for KPZ height - droplet initial condition - flat initial condition - KPZ in half space P. Calabrese, PLD, A. Rosso EPL (2010) P. Calabrese, M. Kormos, PLD, EPL (2014) P. Calabrese, PLD, PRL (2011) J. Stat. Mech. P06001 (2012) T. Gueudre, PLD, EPL (2012). - Integrable directed polymer models on square lattice T. Thiery, PLD, J.Stat. Mech. P10018 (2014) and arxiv Non-crossing probability of directed polymers A. De Luca, PLD, arxiv
3 Outline: - KPZ equation, KPZ class, random matrices,tracy Widom distributions. - solving KPZ at any time by mapping to directed paths then using (imaginary time) quantum mechanics attractive bose gas (integrable) => large time TW distrib. for KPZ height - droplet initial condition - flat initial condition - KPZ in half space P. Calabrese, PLD, A. Rosso EPL (2010) P. Calabrese, PLD, PRL (2011) J. Stat. Mech. P06001 (2012) T. Gueudre, PLD, EPL (2012). - Integrable directed polymer models on square lattice T. Thiery, PLD, J.Stat. Mech. P10018 (2014) and arxiv Non-crossing probability of directed polymers - other works/perspectives: P. Calabrese, M. Kormos, PLD, EPL (2014) A. De Luca, PLD, arxiv V. Dotsenko, H. Spohn, Sasamoto (math) Amir, Corwin, Quastel, Borodine,.. also G. Schehr, Reymenik, Ferrari, O Connell,.. not talk about: stationary initial condition T. Inamura, T. Sasamoto PRL 108, (2012) reviews KPZ: I. Corwin, H. Spohn..
4 Kardar Parisi Zhang equation Phys Rev Lett (1986) growth of an interface of height h(x,t) diffusion noise - 1D scaling exponents - P(h=h(x,t)) non gaussian depends on some details of initial condition flat h(x,0) =0 wedge h(x,0) = - w x (droplet) Edwards Wilkinson P(h) gaussian
5 - Turbulent liquid crystals Takeuchi, Sano PRL (2010) droplet flat also reported in: is a random variable - slow combustion of paper J. Maunuksela et al. PRL (1997) - bacterial colony growth Wakita et al. J. Phys. Soc. Japan. 66, 67 (1996) - fronts of chemical reactions S. Atis (2012) - formation of coffee rings via evaporation Yunker et al. PRL (2012)
6 Large N by N random matrices H, with Gaussian independent entries eigenvalues Universality large N : - DOS: semi-circle law 1 (GOE) 2 (GUE) 4 (GSE) H is: real symmetric hermitian symplectic histogram of eigenvalues N= distribution of the largest eigenvalue Tracy Widom (1994)
7 Tracy-Widom distributions (largest eigenvalue of RM) GOE Fredholm determinants GUE Ai(x-E) is eigenfunction E particle linear potential 0.4 Ai(x) x
8 discrete models in KPZ class/exact results - polynuclear growth model (PNG) Prahofer, Spohn, Baik, Rains (2000) - totally asymmetric exclusion process (TASEP) Johansson (1999) step initial data
9 Exact results for height distributions for some discrete models in KPZ class - PNG model droplet IC Baik, Deft, Johansson (1999) GUE Prahofer, Spohn, Ferrari, Sasamoto,.. (2000+) flat IC GOE multi-point correlations Airy processes GUE GOE - similar results for TASEP Johansson (1999),...
10 Exact results for height distributions for some discrete models in KPZ class - PNG model Baik, Deft, Johansson (1999) Prahofer, Spohn, Ferrari, Sasamoto,.. (2000+) flat IC droplet IC GUE GOE multi-point correlations Airy processes GUE GOE - similar results for TASEP Johansson (1999),... question: is KPZ equation in KPZ class?
11 KPZ equation Cole Hopf mapping Continuum Directed paths (polymers) in a random potential Quantum mechanics of bosons (imaginary time) Kardar 87
12 - Droplet (Narrow wedge) KPZ/Continuum DP fixed endpoints Replica Bethe Ansatz (RBA) - P. Calabrese, P. Le Doussal, A. Rosso EPL (2010) - V. Dotsenko, EPL (2010) J Stat Mech P07010 Dotsenko Klumov P03022 (2010). Weakly ASEP - T Sasamoto and H. Spohn PRL (2010) Nucl Phys B (2010) J Stat Phys (2010). - G.Amir, I.Corwin, J.Quastel Comm.Pure.Appl.Math (2011)
13 - Droplet (Narrow wedge) KPZ/Continuum DP fixed endpoints Replica Bethe Ansatz (RBA) - P. Calabrese, P. Le Doussal, A. Rosso EPL (2010) - V. Dotsenko, EPL (2010) J Stat Mech P07010 Dotsenko Klumov P03022 (2010). Weakly ASEP - T Sasamoto and H. Spohn PRL (2010) Nucl Phys B (2010) J Stat Phys (2010). - G.Amir, I.Corwin, J.Quastel Comm.Pure.Appl.Math (2011) - Flat KPZ/Continuum DP one free endpoint (RBA) P. Calabrese, P. Le Doussal, PRL (2011) and J. Stat. Mech. P06001 (2012) ASEP J. Ortmann, J. Quastel and D. Remenik arxiv and arxiv
14 - Droplet (Narrow wedge) KPZ/Continuum DP fixed endpoints Replica Bethe Ansatz (RBA) - P. Calabrese, P. Le Doussal, A. Rosso EPL (2010) - V. Dotsenko, EPL (2010) J Stat Mech P07010 Dotsenko Klumov P03022 (2010). Weakly ASEP - T Sasamoto and H. Spohn PRL (2010) Nucl Phys B (2010) J Stat Phys (2010). - G.Amir, I.Corwin, J.Quastel Comm.Pure.Appl.Math (2011) - Flat KPZ/Continuum DP one free endpoint (RBA) P. Calabrese, P. Le Doussal, PRL (2011) and J. Stat. Mech. P06001 (2012) ASEP J. Ortmann, J. Quastel and D. Remenik arxiv and arxiv Stationary KPZ => Patrik Ferrari s talk
15 Cole Hopf mapping KPZ equation: define: it satisfies: describes directed paths in random potential V(x,t)
16 Feynman Kac
17 initial conditions 1) DP both fixed endpoints KPZ: narrow wedge <=> droplet initial condition h x 2) DP one fixed one free endpoint KPZ: flat initial condition
18 Schematically calculate guess the probability distribution from its integer moments:
19 Quantum mechanics and Replica.. drop the tilde.. Attractive Lieb-Lineger (LL) model (1963)
20 what do we need from quantum mechanics? - KPZ with droplet initial condition eigenstates = fixed endpoint DP partition sum eigen-energies symmetric states = bosons
21 what do we need from quantum mechanics? - KPZ with droplet initial condition eigenstates = fixed endpoint DP partition sum eigen-energies symmetric states = bosons - flat initial condition
22 LL model: n bosons on a ring with local delta attraction
23 LL model: n bosons on a ring with local delta attraction Bethe Ansatz: all (un-normalized) eigenstates are of the form (plane waves + sum over permutations) They are indexed by a set of rapidities
24 LL model: n bosons on a ring with local delta attraction Bethe Ansatz: all (un-normalized) eigenstates are of the form (plane waves + sum over permutations) They are indexed by a set of rapidities which are determined by solving the N coupled Bethe equations (periodic BC)
25 n bosons+attraction => bound states Bethe equations + large L => rapidities have imaginary parts - ground state = a single bound state of n particules Derrida Brunet 2000 Kardar 87 exponent 1/3
26 n bosons+attraction => bound states Bethe equations + large L => rapidities have imaginary parts - ground state = a single bound state of n particules Derrida Brunet 2000 Kardar 87 exponent 1/3 can it be continued in n? NO! information about the tail of FE distribution
27 n bosons+attraction => bound states Bethe equations + large L => rapidities have imaginary parts - ground state = a single bound state of n particules Derrida Brunet 2000 Kardar 87 - all eigenstates are: need to sum over all eigenstates! exponent 1/3 All possible partitions of n into ns strings each with mj particles and momentum kj
28 Integer moments of partition sum: fixed endpoints (droplet IC) norm of states: Calabrese-Caux (2007)
29 how to get P( ln Z) i.e. P(h)? random variable expected O(1) introduce generating function of moments g(x): so that at large time:
30 how to get P( ln Z) i.e. P(h)? random variable expected O(1) introduce generating function of moments g(x): what we aim to calculate= Laplace transform of P(Z) so that at large time: what we actually study
31 reorganize sum over number of strings
32 reorganize sum over number of strings Airy trick double Cauchy formula
33 Results: 1) g(x) is a Fredholm determinant at any time t by an equivalent definition of a Fredholm determinant
34 Results: 1) g(x) is a Fredholm determinant at any time t by an equivalent definition of a Fredholm determinant Airy function identity 2) large time limit g(x)= GUE-Tracy-Widom distribution
35 P. Calabrese, P. Le Doussal, (2011) needed: 1) g(s=-x) is a Fredholm Pfaffian at any time t
36 P. Calabrese, P. Le Doussal, (2011) needed: 1) g(s=-x) is a Fredholm Pfaffian at any time t 2) large time limit
37 Fredholm Pfaffian Kernel at any time t
38 Fredholm Pfaffian Kernel at any time t large time limit
39 how to calculate first method: flat as limit of half-flat (wedge)
40 how to calculate first method: flat as limit of half-flat (wedge) miracle!
41 how to calculate first method: flat as limit of half-flat (wedge) miracle! strings:
42 in double limit pairing of string momenta and pfaffian structure emerges
43 second method: calculate: use Bethe equations: is the overlap with uniform state => integral vanishes for generic state oberve: requires pairs opposite rapidities Can be seen as interaction quench in! Lieb-Liniger model with initial state BEC (c=0) de Nardis et al., arxiv overlap is non zero only for parity invariant states Brockmann, arxiv P. Calabrese, P. Le Doussal, arxiv large L limit, overlap for strings partially recovers the moments Z^n for flat
44 Summary: we found for droplet initial conditions at large time has the same distribution as the largest eigenvalue of the GUE for flat initial conditions similar (more involved) at large time has the same distribution as the largest eigenvalue of the GOE in addition: g(x) for all times => P(h) at all t (inverse LT) GSE? decribes full crossover from Edwards Wilkinson to KPZ is crossover time scale large for weak noise, large diffusivity
45 Summary: for droplet initial conditions at large time has the same distribution as the largest eigenvalue of the GUE for flat initial conditions similar (more involved) at large time has the same distribution as the largest eigenvalue of the GOE in addition: g(x) for all times => P(h) at all t (inverse LT) GSE? KPZ in half-space decribes full crossover from Edwards Wilkinson to KPZ is crossover time scale
46 DP near a wall = KPZ equation in half space T. Gueudre, P. Le Doussal, EPL (2012) fixed
47 DP near a wall = KPZ equation in half space T. Gueudre, P. Le Doussal, EPL (2012) fixed distributed as Gaussian Symplectic Ensemble
48 Integrable directed polymer (DP) on square lattice J - log-gamma DP on-site weights inverse Gamma distribution Seppalainen (2012) Brunet COSZ(2011) BCR(2013), Thiery, PLD(2014) t I,x
49 Integrable directed polymer (DP) on square lattice J - log-gamma DP on-site weights t inverse Gamma distribution Seppalainen (2012) Brunet COSZ(2011) BCR(2013), Thiery, PLD(2014) - Strict-Weak DP Gamma distribution Corwin,Seppalainen,Shen(2014) O Connell,Ortmann(2014) v w u I,x - Beta DP Beta distribution Barraquand,Corwin(2014)
50 Integrable directed polymer (DP) on square lattice J - log-gamma DP on-site weights t inverse Gamma distribution Seppalainen (2012) Brunet COSZ(2011) BCR(2013), Thiery, PLD(2014) - Strict-Weak DP Gamma distribution Corwin,Seppalainen,Shen(2014) O Connell,Ortmann(2014) v w u I,x - Beta DP Beta distribution Barraquand,Corwin(2014) - Inverse-Beta DP inverse-beta distribution T. Thiery, PLD(2015)
51 Integrable directed polymer models on square lattice T. Thiery, PLD(2015) What do they have in common?
52 => Transfer matrix contains all integer moments
53 Bethe Ansatz => Transfer matrix contains all integer moments
54 Bethe Ansatz => Transfer matrix contains all integer moments A. Povolotsky (2013) zero-range process quantum binomial formula condition for integrability T. Thiery, PLD(2015)
55 Inverse-Beta polymer: main results via Bethe Ansatz T. Thiery, PLD(2015) - Laplace-transform representations conjecture contains log-gamma polymer - large time optimal angle - zero temperature limit exponential-bernoulli bond energies exponential site energies recovers Johansson s 2000 formula
56 Probability that 2 directed polymers in same disorder do not cross Andrea de Luca, PLD, arxiv non-crossing probability Karlin Mc Gregor
57 Probability that 2 directed polymers in same disorder do not cross Andrea De Luca, PLD, arxiv non-crossing probability Karlin Mc Gregor Moments of the non-crossing probability
58 Probability that 2 directed polymers in same disorder do not cross Andrea de Luca, PLD, arxiv non-crossing probability Karlin Mc Gregor Moments of the non-crossing probability Lieb-Liniger with general symmetry (beyond bosons)
59 Nested Bethe ansatz inside irreducible representation of S_n 2-row Young diagram example antisymmetry C-N Yang PRL 19,1312 (1967) auxiliary spin chain auxiliary rapidities => large L: strings again! but not all strings contribute!
60 LL conserved charges
61 LL conserved charges =>
62 LL conserved charges => Introduce GGE partition function =>
63 LL conserved charges => Introduce GGE partition function => We calculated the from the Borodin-Corwin conjecture BC arxiv
64 Results: - first moment: simple from STS!
65 Results: - first moment: simple from STS! - second moment = we find exact relation to average free energy
66
67 Results: - first moment: simple from STS! - second moment = we find exact relation to average free energy - higher moments => conjecture 1 + complicated
68 Results: - first moment: simple from STS! - second moment = we find exact relation to average free energy - higher moments => conjecture 1 + complicated - conjecture 2 => p(t) (sub-)exponentially small for typical environments and for a fraction of environments
69 Results: - first moment: simple from STS! - second moment = we find exact relation to average free energy - higher moments => conjecture 1 + complicated - conjecture 2 => p(t) (sub-)exponentially small for typical environments and for a fraction of environments compare with proba q(t) of single DP not crossing a hard-wall at 0 T. Gueudre, PLD 2012
70 - replica BA method stationary KPZ 2 space points 2 times Perspectives/other works Sasamoto Inamura Prohlac-Spohn (2011), Dotsenko (2013) Dotsenko (2013) Airy process endpoint distribution of DP Dotsenko (2012) Schehr, Quastel et al (2011) - rigorous replica.. q-tasep WASEP Bose gas Borodin, Corwin, Quastel, O Neil,.. avoids moment problem moments as nested contour integrals - sine-gordon FT P. Calabrese, M. Kormos, PLD, EPL (2014) - Lattice directed polymers
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