Scaling limits of inclusion particles. Cristian Giardinà

Transcription

1 Scaling limits of inclusion particles Cristian Giardinà

2 Outline 1. Inclusion process. 2. Models related to inclusion process. 3. Intermezzo: some comments on duality. 4. Scaling limit I: metastability. 5. Scaling limit II: two particles.

3 1. Inclusion process

4 Set up Let S finite set, r x,y 0 jump rates of an irreducible CTRW on S with reversible measure m = (m x ) x S, i.e. m x r x,y = m y r y,x (x, y) S S The reversible inclusion process with parameter k 0 is the Markov process {η(t) : t 0} with state space N S and generator where L f (η) = x,y S S η x,y z = r x,y η x (2k + η y ) [f (η x,y ) f (η)] η x 1 if z = x, η y + 1 if z = y, η z if z {x, y} Introduced in [G., Kurchan, Redig, JMP 07] for k = 1/4.

5 Reversible measure In the gran-canonical ensemble, a family of inhomogeneous product of Negative Binomials with parameters 2k and m x, i.e. µ(η) = 1 Z (φm x ) ηx x S with Z = x S (1 φm x) 2k η x! Γ(η x + 2k) Γ(2k) and 0 < φ < (sup x S m x ) 1 In the canonical ensemble with N particles, the state space is E N = {η N S : η x = N} x S and the unique reversible measure µ N is obtained by conditioning, i.e. µ N (η) = 1 Z N x S m ηx x η x! Γ(η x + 2k) 1 EN (η) Γ(2k)

6 Symmetric case: SIP(k) If r x,y = r y,x then: the random walk reversible measure m is the uniform measure m x = 1 S x S the SIP(k) reversible measure µ is a one-parameter family of i.i.d. Neg Bin (2k,p) with 0 < p < 1 µ(η) = x S 1 p ηx Γ(η x + 2k) (1 p) 2k η x! Γ(2k)

7 2. Two models related to symmetric inclusion process

8 Non-equilibrium statistical mechanics α(2k+η 1 ) η i (2k+η i- 1 ) βη L γη 1 δ(2k+η L ) η i+1 η i- 1 η i 1 i- 1 i i+1 L Adding reservoirs: Bulk: one dimensional chain, nearest neighbor SIP(k) Left: birth/death process with stationary meas. Neg Bin (2k, α γ ) Right: birth/death process with stationary meas. Neg Bin (2k, δ β ) If α γ = δ β If α γ δ β then equilibrium product measure then non-equilibrium measure (long-range correlations) For k = 1/2 it is related to Kipnis-Marchioro-Presutti model [see Carinci, G., Giberti, Redig, JSP 13]

9 Moran process Moran model with population size N, individuals of n types and with symmetric parent-independent mutation at rate θ: a pair of individuals of types x and y are sampled uniformly at random, one dies with probability 1/2 and the other reproduces each individual accumulates mutations at a constant rate θ and his type mutates to any of the others with the same probability. This is the N particle symmetric inclusion process on the complete graph K n with parameter k = L f (η) = x<y n θ n 1 ( ) 2θ η x n 1 + η y [f (η x,y ) f (η)] +η y ( 2θ n 1 + η x see [Carinci, G., Giberti, Redig, SPA 15] ) [f (η y,x ) f (η)]

10 3. Some comments on duality

11 Self-duality Let η(t) and ξ(t) be two independent copies of the SIP(k) process. Consider D(η, ξ) = η x! Γ(2k) (η x x ξ x )! Γ(2k + ξ x ) then E η [D(η(t), ξ)] = E ξ [D(η, ξ(t))] Remark on the use of duality: one can compute n-point correlation functions by using only n-dual walkers. E.g.: In non-equilibrium, if γ = 2k + α and β = 2k + δ then Cov(η x, η y ) = x(l + 1 y) (L + 1) 2 (α δ)2 (2k(L + 1) + 1)

12 Algebraic approach to duality

13 Algebraic approach 1. Write the Markov generator in abstract form, i.e. using the generators of a Lie algebra (typically creation and annihilation operators). 2. Duality is related to a change of representation, i.e. new operators that satisfy the same algebra. Duality functions are the intertwiners. 3. Self-duality is associated to symmetries, i.e. conserved quantities.

14 Duality

15 Self-duality S: symmetry of the generator, i.e. [L, S] = 0, d: trivial self-duality function, D = Sd self-duality function. Indeed LD = LSd = SLd = SdL T = DL T Self-duality is related to the action of a symmetry

16 Construction of Markov generators with algebraic structure and symmetries i) (Lie Algebra): Start from a (representation of 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 the co-product H = (C). v) (Markov generator): Apply a ground state transform (often a similarity transformation) to turn H into a Markov generator L. vi) (Symmetries): S = (X) with X g is a symmetry of H: [H, S] = [ (C), (X)] = ([C, X]) = (0) = 0. [Carinci, G., Redig, Sasamoto, PTRF 16, JSP 16]

17 The method at work: su(1, 1) Lie algebra

18 Algebraic structure of inclusion process L = (K x + Ky + Kx K y + 2KxK o y o + 2k 2) (x,y) E with {K + x, K x, K o x} x S satisfying su(1, 1) Lie algebra [K o x, K ± y ] = ±δ x,y K ± x [K x, K + y ] = 2δ x,y K o x su(1, 1) ferromagnetic quantum spin chain on a graph G = (S, E)

19 step i): representation in terms of matrices A discrete representation of su(1,1) algebra is K + f (n) = (n + 2k) f (n + 1) K f (n) = nf (n 1) K o f (n) = (n + k) f (n) In a canonical base K + = 0 2k... 2k K = K o = k 0 k k

20 step ii): Casimir element For the su(1, 1) algebra the Casimir is C = 1 2 (K K + + K + K ) (K 0 ) 2 C is in the center of the algebra: [C, K + ] = [C, K ] = [C, K o ] = 0 Cf (n) = k(1 k)f (n)

21 step iii): Co-product The co-product is a morphism that turns the algebra into a bialgebra: : su(1, 1) su(1, 1) su(1, 1) and conserves the commutations relations [ (K o ), (K ± )] = ± (K ± ) [ (K ), (K + )] = 2 (K o ) For classical Lie-algebras the co-product is just the symmetric tensor product with the identity (X) = X X := X 1 + X 2

22 (C) = 1 2 step iv): Quantum Hamiltonian ( ) ( ) 2 (K ) (K + ) + (K + ) (K ) (K 0 ) = K 1 K K + 1 K 2 2K o 1 K o 2 + C 1 + C 2 = su(1, 1) Heisenberg ferromagnet + diagonal

23 step v): Markov generator There is no need of a ground state transformation. In this discrete representation where (C) = (L SIP(k) 1,2 ) + 2k(1 2k) L SIP(k) 1,2 f (η 1, η 2 ) = η 1 (η 2 + 2k) [f (η 1 1, η 2 + 1) f (η 1, η 2 )] + η 2 (η 1 + 2k) [f (η 1 + 1, η 2 1) f (η 1, η 2 )] is the generator of the Symmetric Inclusion Process SIP(k).

24 step vi): symmetries As a consequence of the construction, (K α ) with α {+,, o} are symmetries of the process: [(L SIP(k) 1,2 ), K o 1 + K o 2 ] = 0 [(L SIP(k) 1,2 ), K K + 2 ] = 0 [(L SIP(k) 1,2 ), K 1 + K 2 ] = 0

25 Proof self-duality SIP(k) Reversible measure is product of Negative Binomial (2k, p) µ(η) = x S 1 p ηx Γ(2k + η x ) (1 p) 2k η x! Γ(2k) Trivial (i.e. diagonal) self-duality function from reversible measure Symmetry d(η, ξ) = 1 µ(η) δ η,ξ exp{ S 1 (K + )} = exp{ x S K + x }

26 Duality & orthogonal polynomials [Franceschini, G., arxiv: ] [Redig, Sau, arxiv: ]

27 Dualities with orthogonal polynomials Inclusion Process Meixner polynomials Exclusion Process Krawtchouk polynomials Independent walkers Charlier polynomials Brownian momentum process Hermite polynomials Lf (v) = ( ) 2 v x v y f (v) v y v x (x,y) E Brownian energy process Laguerre polynomials Lf (z) = [ ( z x z y z x (x,y) E z y ) 2 ( + 2k(z x z y ) z x z y ) ] f (z)

28 4. Scaling limit I: metastability

29 Infinite population limit, fixed k Proposition: Let {η (N) (t) : t 0} be the SIP(k) process on a graph G = (S, E) initialized with N particles [ L SIP(k) ( ) f (η) = η x 2k + ηy [f (η x,y ) f (η)] (x,y) E ] +η y (2k + η x ) [f (η y,x ) f (η)] The process {z (N) (t) : t 0} defined by z (N) (t) = η(n) (t) N converges in the limit N to the BEP(k) process {z(t) : t 0} initialized from energy 1 L BEP(k) = [ ( z x z y ) 2 ( + 2k(z x z y ) ) ] z x z y z x z y (x,y) E Proof: Taylor expansion + Trotter-Kurtz theorem

30 Condensation Proposition: Consider a parameter k N = k(n) and define d N = 2k N. Suppose d N log N 0 as N. Then where and Moreover lim µ N(η x ) = 1 N S η x z = { N if z = x, 0 if z x x S S = argmax{m(x) : x S} lim N N d N Z N = S Proof: Consequence of Stirling s approximation, essentially proved in [Grosskinsky, Redig, Vafayi, 11].

31 Movement of the condensate Theorem (Bianchi, Dommers, G., 2016). Suppose d N log N 0 as N and that η(0) = η x for some x S. For A E N, let τ A = inf{t 0 : η(t) A}. Then 1. Average time E η x (τ { {y S,y x} ηy }) = 1 1 y S r (1 + o(1)),y x x,y d N 2. Scaling limit X N (t) = z1 {η(t)=η z } z S X N (t/d N ) X(t) weakly as N where X(t) is the Markov process on S with X(0) = x and generator Lf (y) = z S r y,z [f (z) f (y)]

32 Comments In the symmetric case S = S, item 2. recovers the result by [Grosskinsky, Redig, Vafayi 13] Comparison to zero-range process [Beltrán, Landim 12]: Condensation if rates for a particle to move from x to y is r x,y ( ηx η x 1) α for α > 2 Condensate consists of at least N l N particles, l N = o(n); metastable states are equally probable. At time scale t N α+1 the condensate moves from x S to y S at rate proportional to cap(x, y), the capacity of the random walker between x and y.

33 Proof: key ingredients For F : E N R let D N be Dirichlet form D N (F) = 1 2 x,y S ( ) µ N (η) η x dn + η y rx,y [F(η x,y ) F(η)] 2 η E N For two disjoint subsets A, B E N the capacity between A and B can be computed using Dirichlet variational principle where Cap N (A, B) = inf{d N (F) : F F N (A, B)} F N (A, B) = {F : F(η) = 1 for all η A and F(η) = 0 for all η B}.

34 Proof: key ingredients (cont d) The unique minimizer of the Dirichlet principle is the equilibrium potential, i.e., the harmonic function h A,B that solves the Dirichlet problem Lh(η) = 0, if η / A B, h(η) = 1, if η A, h(η) = 0, if η B. It can be easily checked that h A,B (η) = P η (τ A < τ B ). Capacities are related to the mean hitting time between sets [Bovier, Eckhoff, Gayrard, Klein, 01 04] E νa,b (τ B ) = µ N(h A,B ) Cap N (A, B)

35 Proof: key ingredients (cont d) Potential theory ideas and martingale methods can be combined in order to prove the scaling limit of suitably speeded-up processes [Beltrán, Landim, 10 15]. Find a sequence (θ N, N 1) of positive numbers, such that, for any x, y S, x y, the following limit exists p(x, y) := lim θ Np N (η x, η y ) N where p N (η x, η y ) are the jump rates of the original process (θ N ) provides the time-scale to be used in the scaling limit (p(x, y)) x,y S identifies the limiting dynamics.

36 Lemma µ N (η x )p N (η x, η y ) = 1 2 Proof: key ingredients (cont d) Cap N η x, + Cap N η y, z S,z x z S,z y Cap N {η x, η y }, η z η z z S,z {x,y} η z

37 Proof: key ingredients (cont d) Proposition: Let S 1 S and S 2 = S \ S. 1 Then, for d N log N 0 as N, 1 lim Cap N d N η x, η y = 1 r x,y N S x S 1 y S 2 x S 1 y S 2 Combining Lemma and Proposition it follows lim N 1 d N p N (η x, η y ) = r x,y

38 Proof: key ingredients (cont d) Lower bound by restricting the Dirichlet form to suitable subset of E N. Let F s.t. F(η x ) = 1 x S 1 and F(η y ) = 0 y S 2 D N (F) = 1 2 x,y S x S 1 y S 2 = x S 1 y S 2 d N S µ N (η) η x (d N + η y ) r x,y [F (η x,y ) F(η)] 2 η E N µ N (η) η x (d N + η y ) [F (η x,y ) F (η)] 2 r x,y r x,y η x +η y =N N i=1 µ N (i, N i) i (d N + N i) [G(i 1) G(i)] 2 with G(i) = F(η x = i, η y = N i) r x,y (1 + o(1)) x S 1 y S 2

39 Proof: key ingredients (cont d) Upper bound by constructing suitable test function F. Good guess inside tubes η x + η y = N is F(η) η x /N by construction particle moving from x S 1 to y S 2 give correct contribution unlikely to be in a configuration with particles on three sites/ sites not in S unlikely for a particle to escape from a tube

40 Multiple timescales On the time scale 1/d N condensate jumps between site of S. If induced random walk on S is not irreducible, condensate jumps between connected components on longer time scales. Conjecture: if graph distance = 2 then second timescale N d 2 N if graph distance 3 then third timescale N2 d 3 N We prove this when the graph is a line with S = {1,..., L} S = {1, L} r x,y 0 iff x y = 1

41 Second time-scale Theorem (Bianchi, Dommers, G., 2016). Suppose that d N log N 0 as N and η x (0) = N for some x S. Then for one-dimensional system with L = 3 X N (tn/dn 2 ) X(t) weakly as N where X(t) is the Markov process on S = {1, 3} with X(0) = x and transition rates ( 1 p(1, 3) = p(3, 1) = + 1 ) 1 1 r 1,2 r 3,2 1 m 2

42 Third time scale Theorem (Bianchi, Dommers, G., 2016). Suppose that d N log N 0 as N, d N decays subexponentially and η x (0) = N for some x S. Then for one-dimensional system with L 4 there exists constants 0 < C 1 C 2 < such that C 1 lim inf N dn 3 N 2 E η 1[τ dn 3 ηl] lim sup N N 2 E η 1[τ η L] C 2 Conjectured transition rates of time-rescaled process: p(1, L) = p(l, 1) = 3 ( L 2 i=2 ) 1 (1 m i )(1 m i+1 ) m i r i,i+1

43 5. Scaling limit II: two particles on Z

44 One particle Let x(t) denotes the position at time t of a SIP(k) particle on Z: [ ] Lf (x) = 2k f (x + 1) + f (x 1) 2f (x) Given a scaling paramater ɛ > 0 and fixed σ > 0 and 2k ɛ = ɛ σ. Let X ɛ (t) := ɛ x(ɛ 3 t) Then X ɛ (t) X(t) as ɛ 0, with X(t) a Brownian motion on R Lf (X) = 1 σ f (X)

45 Two particles Let x 1 (t) and x 2 (t) denotes the position at time t of two SIP(k) particles (arbitrary but fixed labeling) on Z. Introduce distance and sum coordinates by w(t) := x 2 (t) x 1 (t) u(t) := x 1 (t) + x 2 (t) By definition, the distance and sum coordinates are not depending on the chosen labeling of particles.

46 Distance and sum coordinates The distance process w(t) is autonomous and it evolves as a CTRW on N reflected at 0 and with inhomogeneity in 1 [ ] [L d f ](w) = 1 {w=0} 8k f (w + 1) f (w) ) + 1 {w=1} (4k[f (w + 1) f (w)] + (4k + 2)[f (w 1) f (w)] [ ] + 1 {w 2} 4k f (w + 1) + f (w 1) 2f (w) The sum coordinate u(t), conditionally on {w(t), t 0}, evolves as a CTRW on Z with a defect in w = 1 [ ] [L s f ](u) = 1 {w 1} 4k f (u + 1) + f (u 1) 2f (u) [ ] + 1 {w=1} (4k + 1) f (u + 1) + f (u 1) 2f (u)

47 Fourier-Laplace transform Theorem (Carinci, G., Redig 2017+). Given a scaling paramater ɛ > 0 and fixed σ > 0, consider two SIP(k ɛ ) particles on Z with 2k ɛ = ɛ σ. Let and initial values Then where Ψ (σ) W lim ɛ 0 0 c (σ) κ,λ = U ɛ (t) := ɛ u(ɛ 3 t) 2 W ɛ (t) := ɛ w(ɛ 3 t) 2. ɛu U := lim ɛ 0 2 ɛw W := lim 2 ɛ 0 [ E u,w e i(κ(uɛ(t) U)+mWɛ(t))] e λt dt = Ψ (σ) (κ, m, λ) [ ] (κ, m, λ) = c(σ) κ,λ ΨR W (κ, m, λ) + 1 c (σ) κ,λ κ 2 +2λ κ 2 +2λ+ σ 2 (κ 2 +λ) W Ψ A W (κ, m, λ)

48 Fourier-Laplace transform: σ 0 lim σ 0 Ψ(σ) W (κ, m, λ) = ΨR W (κ, m, λ) Ψ R x (κ, m, λ) := 0 E x [e i(κb t +mb R t ) ] e λt dt {B R t : t 0} Brownian motion on R + reflected at 0 and started from x 0 {B t : t 0} independent standard Brownian motion

49 Fourier-Laplace transform: σ lim σ Ψ(σ) W (κ, m, λ) = ΨA W (κ, m, λ) Ψ A x (κ, m, λ) := 0 E x [e i(κz t +mb A t )] e λt dt {B A t : t 0} Brownian motion on R + absorbed in 0 and started from x 0 Let τ be the absorption time of B A t. Conditionally on τ, Z t := B (1) t τ + B(2) 2(t τ) 1 {t τ} where B (1) t and B (2) t are two standard independent Brownian motions.

50 Fourier-Laplace transform: 0 < σ <, distance coordinate Ψ (σ) W (0, m, λ) = ΨS W (m, λ) Ψ S x (m, λ) := 0 E x [e imbs t ] e λt dt : t 0} sticky Brownian motion on R + with stickiness parameter σ 2 and started from x 0 {B S t Namely B S t = x + B(s(t)) where s 1 (t) = t + σ 2 L(t) L(t) local time of a standard Brownian motion at the origin

51 Perspectives Inclusion process is a novel interacting particle system with mathematical structure of exactly solvable model (e.g. duality) integrability? Condensation regime (infinite population limit) new features (i.e. multiple timescales) compared to other condensing systems, such as zero-range process conjecture: three timescales as found in the 1D setting? thermodynamic limit, coarsening, non-reversible dynamics? Condensation regime (diffusive limit, finite number of particles) two-particles problem: the distance coordinate is sticky BM n-particle dynamics?