Averaging principle and Shape Theorem for growth with memory

Transcription

1 Averaging principle and Shape Theorem for growth with memory Amir Dembo (Stanford) Paris, June 19, 2018 Joint work with Ruojun Huang, Pablo Groisman, and Vladas Sidoravicius

2 Laplacian growth & motion in random media Random growth processes arise in many physical and biological phenomena, going back to Eden (61 )/fpp (Hammersley-Welsh 65 ). Math challenge to understand their evolution and pattern formation. Laplacian growth models: growth at each portion of the boundary determined by the harmonic measure of the boundary from a source. Ex.: Diffusion Limited Aggregation dla (Witten-Sander 81 ), Dielectric Breakdown Model (dbl, Niemeyer et. al. 83 ), idla (Meakin-Deuthch 86 ; Diaconis-Fulton 91 ), hl (Hastings-Levitov 98 ); (see Miller-Sheffield 13 ). Also Abelian sandpiles (Bak et. al. 87 ) & Rotor aggregation (see Levine-Peres 17 ). Related models of motion in random media: Once-Reinforced Random Walk orrw (Davis 90 ), Origin-Excited Random Walk oerw (Kozma 06 ). Amir Dembo Random growth, Shape Theorem 1 / 16

3 Excited walk towards the center Excited random walk: Benjamini-Wilson (04 ), Kosygina-Zerner (12 ). At first visit by rw to a vertex, it gets a (one-time) drift in a fixed direction (e.g. e 1 ), on subsequent visits to vertex perform srw. Simplified model of orrw, where drift direction not fixed (one-time edge/vertex conductance increase a). Kozma (06 ) studies oerw with normalized drift at v direction: E[X t+1 X t F t ] = δ v 2 v if X t = v is first visit of v Z n. Proving recurrence n 1, δ > 0. Kozma (07, 13 ): Conjectured Shape Theorem for most oerw-s. Sidoravicius: same for orrw if reinforcement strength a > 0 large. Having a bulky limit shape connected to the conjectural recurrence of such random walks (confined but not too trapped within a small region). By Shape Theorem get excursion probabilities and deduce recurrence. Amir Dembo Random growth, Shape Theorem 2 / 16

4 Simulations of orrw Vertex orrw on Z 2. Reinforcement strength a = 2 (left), a = 3 (middle), a = 100 (right) in a box of size Color proportional to of vertex first visit time. Amir Dembo Random growth, Shape Theorem 3 / 16

5 Simulations of oerw oerw on Z 2 with different excitation rules. L: move one unit towards the origin in a coordinate chosen with probability proportional to its absolute value. M: move one unit towards the origin in the direction of the coordinate with largest absolute value. R: move one unit towards the origin in each coordinate. Sites colored according to the first visit time by the walk. Amir Dembo Random growth, Shape Theorem 4 / 16

6 Shape Theorems: idla and generalizations idla (Lawler-Bramson-Griffeath 92 ): Each step runs srw till exits current range R(t) Z n. Then t t + 1 & particle back to the origin. η > 0, a.s. B(o, (1 η)t) R(c n t n ) B(o, (1 + η)t) t large. Uniform idla (Benjamini-DuminilCopin-Kozma-Lucas 17 ): Upon exiting R(t) jump to a uniformly chosen v R(t). Same conclusion as for idla. idla with finitely fixed sources & particles moving simultaneously at rate ɛ 1 (Gravner-Quastel 00 ): particles empirical density on (ɛz) n converges weakly as ɛ 0 to solutions of a Stefan free boundary problem. Amir Dembo Random growth, Shape Theorem 5 / 16

7 A random growth model in R n We construct a simplified growth model in R n, n 2, to bypass some technical difficulties of the lattice models. Family of domain processes (D ɛ t) t 0 R n with scale & Poisson rate parameter ɛ (0, 1]. Our rules keep domains star-shaped and compact, so represented by boundary R ɛ t : S n 1 R +, a pure jump (slow) process evolving by adding a small bump of approximate volume ɛ at random points on the boundary. Growth location driven by a (fast) particle process (x ɛ t) t 0 via hitting probability density F (R ɛ t, x ɛ t, ) on S n 1, specifying law of the angle ξ t where a bump is to be added. Mapping H(R ɛ t, ξ t ) says to where particle is transported upon hitting the boundary at angle ξ t. F (r, x, ) : C(S n 1 ) R n L 2 (S n 1 ), H(r, z) : C(S n 1 ) S n 1 R n. Example: F (r, x, ) Poisson kernel of r, H(r, z) = αr(z)z, α [0, 1). Amir Dembo Random growth, Shape Theorem 6 / 16

8 Spherical approximate identity & small bump on S n 1 sai {g η ( z, )} η>0 such that: g η C([ 1, 1], R + ), 1 g η = 1, f g η 2 f 2 & f g η f 2 0 as η 0 (where denotes spherical convolution). E.g. g η (s) = cη (n 1) φ ( 1 1 s η 2 ) for φ C([ 1, 1], R+ ), φ( 1) = 0. For a bump centered at angle ξ add damped sai ɛ 1/n η n 1 g η ( ξ, ) with η(ɛ, r, x) = ɛ 1/n yr,x 1/(n 1). Such bump has height O(ɛ 1/n ) & support on the spherical cap of Euclidean radius 2η around ξ L: g η (s) at different η. R: Adding g η ( z, ) to S 2 ; z = (0, 0, 1). Amir Dembo Random growth, Shape Theorem 7 / 16

9 Our random growth model y r,x := ω n S n 1 r(θ) n 1 F (r, x, θ)dσ(θ), ω n = σ(s n 1 ) = Each bump adds on average volume ɛ + o(ɛ) to D ɛ t. Hitting kernel: F (r, x, ) : C(S n 1 ) R n L 2 (S n 1 ). Transportation function: H(r, z) : C(S n 1 ) S n 1 R n. (Rt, ɛ x ɛ t) t 0 updates at arrival times {Ti ɛ} of a rate ɛ 1 Poisson process. At each t = Ti ɛ, conditional on F t := σ(rɛ s, x ɛ s, ξ s : s t ), let R ɛ t(θ) = R ɛ t (θ) + d ξ t F (R ɛ t, x ɛ t, ), xɛ ɛ 1/n η(ɛ,r ɛ t,x ɛ t ) n 1 {}}{ ɛ y R ɛ t,xɛ t g η(ɛ,r ɛ t,xɛ t ) ( ξ t, θ ), θ S n 1, t = H(R ɛ t, ξ t). Amir Dembo Random growth, Shape Theorem 8 / 16

10 Simulations of the random growth model L: H(r, ξ) = (r(ξ) 1) + ξ. M: H(r, ξ) = (r(ξ) ξ ξ 2 ) + ξ. R: H(r, ξ) = (r(ξ) ξ 1 ξ 2 ) + ξ. F (r, x, ) harmonic measure on r (from x). Snapshots t = 0.2 k 2, 0 k 10, for ɛ = 10 4, φ = 1 [0,1], c = 20, Linear in k evolution O( t) asymptotic diameter growth. Small t spherical shape (H = o, idla-like). Large t R ɛ t feels the geometry H-dependent limit shape (sphere, square, diamond; similar to different excitation for oerw). Amir Dembo Random growth, Shape Theorem 9 / 16

11 Frozen domain & limiting ODE Domain Rt ɛ r frozen = (x ɛ,r t ) t 0 Markov process. Assume r C(S n 1 ), process {x 1,r t } has a unique invariant law ν r. Then, limiting infinite-dimensional ode for R ɛ t is r t (θ)= r 0 (θ) + t 0 b( rs )(θ)ds, θ S n 1 (ode) b(r)(θ) := R n b(r, x)(θ)dν r (x), b(r, x)( ) := ω n y r,x F (r, x, ) mg decomposition of {R ɛ t} has bv-term (drift) b ɛ (r, x) := b(r, x) g η(ɛ,r,x) b(r, x) as ɛ 0 ( η 0), and mg term of O(ɛ) quadratic variation. Leb(r t ) = Leb(r 0 ) + t E[Leb(D ɛ t) F t ] Leb(D ɛ t ) ɛ in line with at each t = T ɛ i Amir Dembo Random growth, Shape Theorem 10 / 16

12 Assumptions on F & H Assumption (L) A 1 (a) := { r C(S n 1 ) : inf θ r(θ) a, r 2 a 1} A(a) := { (r, x) D(F ) : r A 1 (a), x Im(H(r, )) } K = K(a) < so for all (r, x), (r, x ) A(a), z, z S n 1 F (r, x, ) F (r, x, ) 2 K ( r r 2 + x x ), (LF ) H(r, z) H(r, z ) K ( r r 2 + z z ), Assumption (E) b(r) b(r ) 2 K r r 2. (LH) (L b) r C(S n 1 ), the invariant probability law ν r of {x 1,r t } exists & unique. [ lim sup 1 t [b(r, x 1,r s ) t t b(r)]ds ] 2 = 0. 2 E (r,x 1,r 0 ) A(a) 0 For (E) suffices to have uniform minorization of jump kernel P r of {x 1,r T i } inf (r,x) A(a) {(P r) n0 (x, )} m( ). Amir Dembo Random growth, Shape Theorem 11 / 16

13 Hydrodynamic limit by Averaging Principle Theorem (DGHS,18 ) (a) Assume (L) & (E). Fix R0 ɛ = r 0 C(S n 1 ). Then, ( ) lim P sup Rt ɛ r t 2 > ι = 0, T, ι, δ > 0 ( ) ɛ 0 0 t T σ ɛ (δ) with F t -stopping time σ ɛ (δ) := inf{t 0 : min{f (R ɛ θ t, x ɛ t, θ)} < δ}. (b) No σ ɛ (δ) in ( ) when inf{f (r, x, θ) : (r, x) A(a), θ S n 1 } > 0. Proof: As ɛ 0, dynamic of {x ɛ t} near equilibrium at a time scale where {R ɛ t} does not change macroscopically, namely we have here an averaging principle. Amir Dembo Random growth, Shape Theorem 12 / 16

14 Towards shape theorem: scale invariance Assumption (I) c > 0 if (r, x) D(F ) then (cr, cx) D(F ) and F (r, x, ) = F (cr, cx, ) ch(r, ) = H(cr, ) Example: (I) holds for F (r, x, ) Poisson kernel at D (from x), r = D; & H(r, z) = αr(z)z for z S n 1, fixed α (0, 1). Proposition (coupling) Under Assumption (I), there exists coupling with (Rt, ɛ x ɛ t) = (ɛ 1/n Rt/ɛ 1, ɛ1/n x 1 t/ɛ ) for all t > 0, whenever holding for t = 0. Amir Dembo Random growth, Shape Theorem 13 / 16

15 Candidate limiting shapes ψ C(S n 1 ) invariant (shape) for r t (θ) = r 0 (θ) + t 0 b( rs )(θ)ds, θ S n 1 (ode) r 0 = ψ yields r t = c t ψ in (ode); (c t ) n = Leb(ψ) + t. Recall: b(r)(θ) = ωn R n y 1 r,xf (r, x, θ)dν r (x) (I) ν cr (c ) = ν r ( ), y cr,cx = c n 1 y r,x b(cr) = c (n 1) b(r) Under (I) shape ψ is invariant b(ψ)(θ) = 1 nψ(θ) θ Sn 1 Ex: F (r, x, ) Poisson kernel & H(r, z) = α(z)r(z)z ψ = c (Euclidean ball) invariant α(z) = α (constant). Amir Dembo Random growth, Shape Theorem 14 / 16

16 Shape theorem Theorem (DGHS 18 ) Assume (I) and hydrodynamic limit ( ) lim sup Rt ɛ r t 2 > ι = 0, ɛ 0 T, ι, δ > 0 ( ) 0 t T If ψ invariant for ode (wlog Leb(ψ) = 1), then T, c, ι > 0, ( lim P sup (N(c + s)) 1/n R sn ψ N 2 > ι ) R 0 = (Nc) 1/n ψ = 0 1 s T While F (r, x, ) F (r, x, ) 2 K(a) ( r r 2 ) fails for Poisson kernel F (r, x, ), this is resolved upon regularizing F. (LF ) Amir Dembo Random growth, Shape Theorem 15 / 16

17 Thank you!

18 Generator & mg decomposition The generator of (Rt, ɛ x ɛ t) t 0 is ( L ɛ f ) (r, x) = 1 [ ( r + ɛ ) ] g η(ɛ,r,x) ( ξ, ), H(r, ξ) F (r, x, ξ)dσ(ξ) f(r, x). ɛ y r,x S n 1 f Taking f(r, x) = r(θ) at fixed θ S n 1 yields R ɛ t(θ) = R ɛ 0(θ) + where Σ ɛ t(θ) a (small) mg. t Taking f(r, x) = x e i, i = 1,..., n, yields for some R n -valued mg M ɛ t. t 0 b ɛ (R ɛ s, x ɛ s)(θ)ds + Σ ɛ t(θ) x ɛ t = x ɛ 0 + ɛ 1 h(rs, ɛ x ɛ s)ds + Mt ɛ 0 h(r, x) := H(r, ξ)f (r, x, ξ)dσ(ξ) x, S n 1 Amir Dembo Random growth, Shape Theorem 16 / 16