An algebraic approach to stochastic duality. Cristian Giardinà

Transcription

1 An algebraic approach to stochastic duality Cristian Giardinà RAQIS18, Annecy 14 September 2018

2 Collaboration with Gioia Carinci (Delft) Chiara Franceschini (Modena) Claudio Giberti (Modena) Jorge Kurchan (ENS Paris) Frank Redig (Delft) Tomohiro Sasamoto (Tokyo Tech)

3 Outline Introduction Lie algebraic approach to duality theory su q (1, 1) algebra Applications

4 Introduction

5 Non-equilibrium in 1d: particle transport 1 q asymmetry 0 N 1 1 Ρ density reservoirs 0 N Ρ

6 Non-equilibrium in 1d: particle transport 1 q asymmetry 0 N 1 1 Ρ density reservoirs 0 N Ρ j N 1 1 j N 0 N current reservoirs Carinci, De Masi, G., Presutti (2016)

7 Non-equilibrium in 1d: energy transport Fourier law J = κ T KMP model (1982) Energies at every site: z = (z 1,..., z N ) R N + L KMP f (z) = N 1 [ ] dp f (z 1,..., p(z i + z i+1 ), (1 p)(z i + z i+1 ),..., z N ) f (z) i=1 0 conductivity 0 < κ < ; model solved by duality.

8 Definition Stochastic Duality (η t ) t 0 Markov process on Ω with generator L, (ξ t ) t 0 Markov process on Ω dual with generator L dual ξ t is dual to η t with duality function D : Ω Ω dual R if t 0 E η (D(η t, ξ)) = E ξ (D(η, ξ t )) (η, ξ) Ω Ω dual η t is self-dual if L dual = L. In terms of generators: LD(, ξ)(η) = L dual D(η, )(ξ)

9 Duality for Markov chain Assume state spaces Ω, Ω dual are countable sets, then the Markov generator L is a matrix L(η, η ) s.t. L(η, η ) 0 if η η, L(η, η ) = 0 η Ω LD(, ξ)(η) = L dual D(η, )(ξ) amounts to LD = DL T dual Indeed L(η, η )D(η, ξ) = L dual (ξ, ξ )D(η, ξ ) η ξ

10 Duality A useful tool interacting particle systems [Spitzer, Ligget] hydrodynamic limit [Presutti, De Masi] KPZ scaling limits [Schütz, Spohn] population genetics [Kingman]... the dual process is simpler: from many to few. Questions how to find a dual process and a duality function? how to construct processes with duality? * E.g.: duality for asymmetric partial exclusion process? * E.g.: what is the right asymmetric version of KMP?

11 Lie algebraic approach to duality theory

12 Algebraic approach Write the Markov generator in abstract form, i.e. as an element of a universal enveloping algebra of a Lie algebra. 1. Duality is related to a change of representation. Duality functions are the intertwiners. 2. Dualities are associated to symmetries. Acting with a symmetry on a duality fct. yields another duality fct. Conversely, the approach can be turned into a constructive method.

13 1. Change of representation

14 Example: Wright-Fisher diffusion and Kingman coalescence Wright-Fisher diffusion (X(t)) t 0 with state space [0, 1] L WF f (x) = 1 2 x(1 x) 2 f x 2 (x) N(t) = number of blocks in the Kingman coalescence at time t 0 (L King f )(n) = n(n 1) (f (n 1) f (n)) 2

15 Duality Wright-Fisher / Kingman The process {X(t)} t 0 with generator L WF and the process {N(t)} t 0 with generator L King are dual on D(x, n) = x n, i.e. E WF x (X(t) n ) = E King n (x N(t) ) Indeed: L WF D(, n)(x) = 1 2 x(1 x) 2 x 2 x n n(n 1) = (x n 1 x n ) 2 n(n 1) = (D(x, n 1) D(x, n)) 2 = L King D(x, )(n)

16 Duality Wright-Fisher / Kingman : algebraic approach Two representations of the Heisenberg algebra: [a, a ] = 1 a = x a e (n) = e (n+1) a = d dx a e (n) = ne (n 1) The abstract element L = 1 2 a (1 a )(a) 2 L = L WF L T = L King in the first representation in the second representation Duality fct. D(x, n) = x n is the intertwiner: xd(x, n) = D(x, n + 1) d D(x, n) = nd(x, n 1) dx

17 2. Symmetries S: symmetry of the original Markov generator, i.e. [L, S] = 0 d: duality function between L and L dual D = Sd is also duality function Indeed LD = LSd = SLd = SdL T dual = DL T dual

18 Cheap self-duality Let µ a reversible measure: µ(η)l(η, ξ) = µ(ξ)l(ξ, η) A cheap (i.e. diagonal) self-duality is d(η, ξ) = 1 µ(η) δ η,ξ Indeed L(η, ξ) µ(ξ) = η L(η, η )d(η, ξ) = ξ L(ξ, ξ )d(η, ξ ) = L(ξ, η) µ(η)

19 Cheap duality Let µ a invariant measure: η µ(η)l(η, ξ) = 0 Let the dual process (ξ t ) t 0 be the time-reversed process of (η t ) t 0 L dual (ξ, ξ ) = µ(ξ) 1 L(ξ, ξ)µ(ξ ) A cheap (i.e. diagonal) duality is d(η, ξ) = 1 µ(η) δ η,ξ L(η, ξ) µ(ξ) Indeed = η L(η, η )d(η, ξ) = ξ L dual (ξ, ξ )d(η, ξ ) = L(η, ξ) µ(ξ)

20 Construction of Markov generators with algebraic structure i) (Lie Algebra): Start from a Lie algebra g. ii) (Casimir): Pick an element in the center of g, e.g. the Casimir C. iii) (Co-product): Consider a co-product : g g g making the algebra a bialgebra and conserving the commutation relations. iv) (Quantum Hamiltonian): Compute H = (C). v) (Symmetries): S = (X) with X g is a symmetry of H: [H, S] = [ (C), (X)] = ([C, X]) = (0) = 0. vi) (Markov generator): Apply a ground state transformation to turn H into a Markov generator L.

21 Quantum su q (1, 1) algebra

22 q-numbers For q (0, 1) and n N 0 introduce the q-number [n] q = qn q n q q 1 Remark: lim q 1 [n] q = n. The first q-number s are: [0] q = 0, [1] q = 1, [2] q = q+q 1, [3] q = q 2 +1+q 2,...

23 Quantum Lie algebra su q (1, 1) For q (0, 1) consider the algebra with generators K +, K, K 0 where [K 0, K ± ] = ±K ±, [K +, K ] = [2K 0 ] q [2K 0 ] q := q2k 0 q 2K 0 q q 1 Irreducible representations are infinite dimensional. E.g., for n N Casimir element In this representation K + e (n) = [n + 2k] q [n + 1] q e (n+1) K e (n) = [n] q [n + 2k 1] q e (n 1) K o e (n) = (n + k) e (n) C = [K o ] q [K o 1] q K + K C e (n) = [k] q [k 1] q e (n) k R +

24 Co-product Co-product : U q (su(1, 1)) U q (su(1, 1)) 2 (K ± ) = K ± q K o + q K o K ± (K o ) = K o K o The co-product is an isomorphism s.t. [ (K o ), (K ± )] = ± (K ± ) [ (K + ), (K )] = [2 (K o )] q From co-associativity ( 1) = (1 ) n : U q (su(1, 1)) U q (su(1, 1)) (n+1), i.e. for n 2 n (K ± ) = n 1 (K ± ) q K o n+1 + q n 1 (K 0 i ) K ± n+1 n (K o ) = n 1 (K o ) n K 0 n+1

25 Quantum Hamiltonian { } (C i ) = q K i 0 K + i K i+1 + K i K + i+1 B i B i+1 q K i+1 0

26 Quantum Hamiltonian { } (C i ) = q K i 0 K + i K i+1 + K i K + i+1 B i B i+1 q K i+1 0 B i B i+1 = (qk + q k )(q k 1 + q (k 1) ) ( ) ) q K 0 2(q q 1 ) 2 i q K i 0 (q K i+1 0 q K i (qk q k )(q k 1 q (k 1) ) ( ) ) q K 0 2(q q 1 ) 2 i + q K i 0 (q K i q K i+1 0

27 Quantum Hamiltonian { } (C i ) = q K i 0 K + i K i+1 + K i K + i+1 B i B i+1 q K i+1 0 H := L 1 (1 (i 1) (C i ) 1 (L i 1) + c q,k 1 L) i=1 c q,k = (q2k q 2k )(q 2k 1 q (2k 1) ) ( s.t. H (q q 1 ) 2 L i=1e (0) i ) = 0

28 Markov processes with su q (1, 1) symmetry

29 Symmetries of H Lemma K ± := K 0 := L i=1 i=1 q K 0 1 q K 0 i 1 K ± i q K 0 i+1... q K 0 L L } 1 {{ 1 } Ki 0 1 } {{ 1 } (i 1) times (L i) times. are symmetries of H. Proof. Let a {+,, 0}, then K a = L 1 (K a 1 ) For L = 2 : [H, K a ] = [ (C 1 ), (K1 a )] = ([C 1, K1 a ]) = (0) = 0 For L > 2 : induction.

30 Lemma Ground state transformation Let H be a matrix with H(η, η ) 0 if η η. Suppose g is a positive ground state, i.e. H g = 0 and g(η) > 0. Let G be the matrix G(η, η ) = g(η)δ η,η. Then is a Markov generator. L = G 1 H G Indeed Therefore L(η, η ) = H(η, η )g(η ) g(η) L(η, η ) 0 if η η η L(η, η ) = 0

31 Exponential symmetries g (0) = L i=1 e(0) i is a ground state, i.e. Hg (0) = 0. For every symmetry [H, S] = 0 another ground state is g = Sg (0). The exponential symmetry with S + = exp q 2(E) = n 0 (E) n [n] q! q n(n 1)/2 E = (L 1) (q K 0 1 ) (L 1) (K + 1 ) gives a positive ground state g = S + g (0) = ( (li ) + 2k 1 Remark l 1,...,l L L i=1 l i lim q 1 S+ = e i K + i = i q e K + i q l i (1 k+2ki) ) e (l i )

32 (1) Asymmetric Inclusion Process: ASIP(q,k) For k R + the interacting particle system ASIP(q, k) on [1, L] Z with state space N L is defined by with L 1 (L ASIP(q,k) f )(η) = (L i,i+1 f )(η) (L i,i+1 f )(η) = q η i η i+1 +(2k 1) [η i ] q [2k + η i+1 ] q (f (η i,i+1 ) f (η)) i=1 + q η i η i+1 (2k 1) [2k + η i ] q [η i+1 ] q (f (η i+1,i ) f (η)) q 1 SIP(k): symmetric inclusion jump right at rate η i (2k + η i+1 ), jump left at rate (2k + η i )η i+1

33 Properties of ASIP(q,k) The ASIP(q, k) on [1, L] Z has a family (labeled by α > 0) of inhomogeneous reversible product measures with marginals ( ) P α (η i = x) = αx x + 2k 1 q 4kix Z i,α x q q 1: the reversible measure is homogeneous and product of Negative Binomials (2k, α)

34 (2) Asymmetric Brownian Energy Process: ABEP(σ, k) For σ > 0, let (η (ɛ) (t)) t 0 be the ASIP(1 ɛσ, k) process initialized with ɛ 1 particles. The scaling limit (weak asymmetry) z i (t) := lim ɛ η (ɛ) ɛ 0 i (t) is the diffusion ABEP(σ, k) with generator L ABEP(σ,k) = L 1 i=1 L i,i+1 L i,i+1 = 1 2σ { (1 e 2σz i )(e 2σz i+1 1) + 2k ( 2 e 2σz i e 2σz ) } ( i σ 2 (1 e 2σz i )(e 2σz i+1 1) ( z i z i+1 ) 2 z i z i+1 ) Remark: L i,i+1 conserves z i (t) + z i+1 (t)

35 Properties of ABEP(σ, k) σ 0 the process is truly asymmetric, i.e. on the 1-d torus it carries a non-zero current. on the half-line it has inhomogeneous reversible product measures (labeld by γ > 4σk) with marginal density σ 0 + µ(dz i ) = 1 Z i,α (1 e 2σz i ) (2k 1) e (4σki+γ)z i dz i L i,i+1 = 2k (z i z i+1 ) ( z i z i+1 ) + z i z i+1 ( z i The reversible measures are given by product of i.i.d. Gamma(2k; γ) µ(dz i ) = 1 γ 2k Γ(2k) z i (2k 1) e γz i dz i z i+1 ) 2

36 (3) KMP(k) process Instantaneous thermalization limit: ( ) L KMP(k) i,j f (z i, z j ):= lim e tlbep(k) i,j 1 f (z i, z j ) = t 1 0 dp ν (k) (p) [f (p(z i + z j ), (1 p)(z i + z j )) f (z i, z j )] Z i, Z j Gamma (2k, θ) i.i.d. = P = Z i Z i + Z j Beta (2k, 2k) ν (k) (p) = p2k 1 (1 p) 2k 1 B(2k, 2k) For k = 1 2 : uniform redistribution, original KMP

37 Asymmetric KMP-like processes AKMP(σ, k) L AKMP(σ,k) i,j f (z i, z j ):= lim = with 1 0 t ( ) e tlabep(σ,k) i,j 1 f (z i, z j ) dp ν (k) σ (p z i + z j ) [f (p(z i + z j ), (1 p)(z i + z j )) f (z i, z j )] ν σ (k) (p E) = 1 {( ) e 2σpE e 2σpE 1 (1 e 2σ(1 p)e)} 2k 1 N σ,k Th-ASIP(k) (n, m) (R q, n + m R q ) with R q a q-deformed Beta-Binomial (n + m, 2k, 2k)

38 Duality relations

39 Duality between ABEP(σ, k) and SIP(k) Theorem [Carinci,G., Redig, Sasamoto (2016)] For every σ (including 0 + ), the process {z(t)} t 0 with generator L ABEP(σ,k) and the process {η(t)} t 0 with generator L SIP(k) are dual on with D(z, ξ) = L i=1 ( Γ(2k) e 2σEi+1(z) e 2σE i (z) Γ(2k + ξ i ) 2σ E i (z) = L z l E L+1 (z) = 0 l=i ) ξi Same duality holds between AKMP(σ, k) and Th-SIP(k)

40 L 1 ( L = i=1 the symmetric case σ = 0 + K + i K i+1 + K i K + i+1 2K i o Ki+1 o + 2k 2) Two representations of the su(1, 1) algebra: K + i e (η i ) = (η i + 2k) e (η i +1) K + i = z i K i e (η i ) = η i e (η i 1) K i = z i z 2 i + 2k zi Ki o e (η i ) = (η i + 4k) e (η i ) Ki o = z i zi + k L = L SIP(k) Γ(2k) Γ(2k + ξ i ) zξ i i Duality fct intertwiner L = L BEP(k)

41 the asymmetric case σ 0 The ABEP(σ, k) can be mapped to BEP(k) via the non-local transformation Equivalently z g(z) g i (z) := e 2σE i+1(z) e 2σE i (z) 2σ L ABEP(σ,k) = C g L BEP(k) C g 1 with (C g f )(z) = (f g)(z) Therefore, despite the asymmetry, the symmetry group of ABEP(σ, k) is the same as for BEP(k), namely su(1, 1). The representation is a non-local conjugation of the differential operator representation.

42 Self-duality of ASIP(q, k) Theorem [Carinci,G., Redig, Sasamoto (2016)] The ASIP(q, k) is self-dual on n D(η, ξ (l 1,...,l n) ) = q 4k m=1 lm n2 (q 2k q 2k ) n n (q 2N lm (η) q 2Nlm+1(η) ) m=1 where ξ (l 1,...,l n) is the configuration with n particles at sites l 1,..., l n and L N i (η) := k=i η k It follows from the explicit knowledge of the reversible measure and from an exponential symmetry.

43 Applications 1. Bulk driven: current 2. Boundary driven: correlation functions

44 Definition Example 1: bulk-driven ABEP(σ, k) The current J i (t) during the time interval [0, t] across the bond (i 1, i) is defined as: J i (t) = E i (z(t)) E i (z(0)) where E i (z) := k i z k Remark: let ξ (i) be the configuration with 1 dual particle: { ξ (i) 1 if m = i m = 0 otherwise then ( ) D(z, ξ (i) e 2σEi+1(z) e 2σE i (z) ) = 4kσ

45 Current of bulk-driven ABEP(σ, k) Using duality between ABEP(σ, k) and SIP(k) E z (e 2σJ i (t) ) = e 4kt n Z e 2σ(En(z) E i (z)) I n i (4kt) I n (t) modified Bessel function. The computation requires a single dual SIP particle, which is a simple symmetric random walk X t jumping at rate 2k P i (X t = n) = e 4kt I n i (4kt)

46 Example 2: Brownian Momentum Process with reservoirs Generator N 1 L = L left + L BMP i,i+1 + L right i=1

47 Example 2: Brownian Momentum Process with reservoirs Generator N 1 L = L left + L BMP i,i+1 + L right i=1 ( L BMP i,i+1 = x i ) 2 x i+1 Bulk x i+1 x i

48 Example 2: Brownian Momentum Process with reservoirs Generator N 1 L = L left + L BMP i,i+1 + L right i=1 ( L BMP i,i+1 = x i ) 2 x i+1 Bulk x i+1 x i L left = T L 2 x 2 1 x 1 x 1 Reservoir

49 Example 2: Brownian Momentum Process with reservoirs Generator N 1 L = L left + L BMP i,i+1 + L right i=1 ( L BMP i,i+1 = x i ) 2 x i+1 Bulk x i+1 x i L left = T L 2 x 2 1 x 1 x 1 Reservoir T L = T R = T : equilibrium Gibbs measure ν T = N i=1 N (0, T ). T L T R : non-equilibrium steady state

50 SIP with absorbing boundaries Configurations ξ = (ξ 0, ξ 1,..., ξ N, ξ N+1 ) Ω dual = N N+2

51 SIP with absorbing boundaries Configurations ξ = (ξ 0, ξ 1,..., ξ N, ξ N+1 ) Ω dual = N N+2 Generator N 1 L dual = L left + L SIP i,i+1 + L right i=1

52 SIP with absorbing boundaries Configurations ξ = (ξ 0, ξ 1,..., ξ N, ξ N+1 ) Ω dual = N N+2 Generator L SIP N 1 L dual = L left + L SIP i,i+1 + L right i=1 N 1 i,i+1 f (ξ) = ξ i (ξ i )[f (ξi,i+1 ) f (ξ)] i=1 Bulk + ξ i+1 (ξ i )[f (ξi+1,i ) f (ξ)]

53 SIP with absorbing boundaries Configurations ξ = (ξ 0, ξ 1,..., ξ N, ξ N+1 ) Ω dual = N N+2 Generator L SIP N 1 L dual = L left + L SIP i,i+1 + L right i=1 N 1 i,i+1 f (ξ) = ξ i (ξ i )[f (ξi,i+1 ) f (ξ)] i=1 Bulk + ξ i+1 (ξ i )[f (ξi+1,i ) f (ξ)] L left f (ξ) = 2ξ 1 (f (ξ 1,0 ) f (ξ)) Boundary

54 Duality and correlation functions with E x [D(x(t), ξ)] = E dual ξ [D(x, ξ(t))] D(x, ξ) = T ξ 0 L ( N i=1 x 2ξ i i Γ( 1 2 ) Γ(ξ i ) ) T ξ N+1 R As a consequence D(x, ξ)µ(dx) = n+m= ξ T a L T b R P ξ(n, m) P ξ (n, m) = P(n particles will exit left, m exit right ξ(0) = ξ) µ stationary measure, ξ = N i=1 ξ i

55 Temperature profile 1d linear chain ξ = (0,..., 0, 1, 0,..., 0) D(x, ξ ) = x 2 i 1 SIP(1) walker (X t ) t 0 with X 0 = i site i ( E x 2 i ) = T L P i (X = 0) + T R P i (X = N + 1) E(x 2 i ) = T L + ( ) TR T L i N + 1 E(J) = E(x 2 i+1 x 2 i ) = T R T L N + 1 Linear profile Fourier s law

56 Energy covariance 1d linear chain If ξ = (0,..., 0, 1, 0,..., 0, 1, 0,..., 0) D(x, ξ ) = x 2 site i site j i xj 2 In the dual process we initialize two SIP walkers (X t, Y t ) t 0 with (X 0, Y 0 ) = (i, j)

57 Inclusion Process with absorbing reservoirs i SIP L j L abs 1,0 f f 1 f 2 1 abs L N

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66 i j x T P T P T T P P E x i j L R L R

67 ( ) ( E xi 2 xj 2 E x 2 i ) Energy covariance ( E x 2 j ) = 2i(N + 1 j) (N + 3)(N + 1) 2 (T R T L ) 2 Remark 1: up to a sign, same covariance as in the boundary driven Symmetric Simple Exclusion Process. Remark 2: Long range correlations lim N Cov(x α 2 N 1 N, x α 2 2 N ) = 2α 1(1 α 2 )(T R T L ) 2

68 Summary Two key ingredients for stochastic duality: symmetries and representation theory Constructive Lie-algebraic approach to duality theory Examples: suq (1, 1) algebra suq (2) algebra gives ASEP(q,j) with duality higher rank algebras give multispecies interacting particle systems with duality Scaling to KPZ universality class of ABEP and ASIP?

69 Some references C. Franceschini, C. Giardinà Stochastic Duality and Orthogonal Polynomials arxiv: G. Carinci, C. Giardinà, F. Redig, T. Sasamoto Asymmetric stochastic transport models with U q(su(1, 1)) symmetry, JSP 163, (2016) G. Carinci, C. Giardinà, F. Redig, T. Sasamoto A generalized Asymmetric Exclusion Process with U q(sl 2 ) stochastic duality, PTRF 166(3), (2016) J. Kuan, A multi-species ASEP(q,j) and q-tazrp with stochastic duality, IMRN 17, (2018) G. Carinci, C. Giardinà, C. Giberti, F. Redig Dualities in population genetics: a fresh look with new dualities SPA 125 (3), (2015) V. Belitsky, G.M. Schütz, Quantum algebra symmetry of the ASEP with second-class particles, JSP 161, (2015).

70 Thank you for your attention