Astonishing Cellular Automata Alexander E. Holroyd UBC and Microsoft

Astonishing Cellular AutomataAlexander E. Holroyd

UBC and Microsoft

Each site of Z2 contains:North-facing car (")or East-facing car (!)or empty space (0).

At odd time steps, each " tries to move one unit North

(succeeds if there is a 0 for it to move into).At even time steps, each ! tries to move one unit East (succeeds if there is a 0 for it to move into).

Biham-Middleton-Levine traffic model (1992)

"!!

0

"!!

1

"!!

2

" !!

3

" !!

4

"!!

5

"!!

6

!!

7

!

8

Random initial configuration:

Parameter p2[0,1].

Each site of Z2 contains:North-facing car (") with probability p/2East-facing car (!) with probability p/2empty space (0) with probability 1 – p

independently for different sites.

Simulation (Mirek’s Cellebration)

Conjecture. 9 pJ2(0,1) s.t.p > pJ: no car moves infinitely often (jammed)p < pJ: all cars move infinitely often

Conjecture. 9 pF 2 (0,1) s.t.p < pF: all cars have asymptotic speed ½ (free

flowing)p > pF: all cars have asymptotic speed < ½

Question. On Z2, pF = pJ ?

Question. On n by n torus, as n!1, 9 intermediate phases:

pF < pJ ? (D’Souza 2005)

Theorem (Angel, Holroyd, Martin 2005). 9 p1 < 1 such that for p > p1 , traffic is jammed (no car moves infinitely often) with probability 1.

In fact, for p > p1, some cars never move...

Only rigorous result (but: Benjamini, Gurel-Gurevich,Izkovsky: 1-dimensional variant)

"!

Easy case: p = 1.Any car is blocked an infinite chain of others:

!! !

""!

Argument does not work for p < 1.Chain will be broken by an empty space.

0You

Proof

Another way for a car to be blocked:

!"

"

"! !

!You Blocked (never moves)


!"

"

"! !


If this never moves,you are blocked


!"

"

"! !


If this does move....


!"

"

"! !



!"

"

"! !



!"

"

"!

!StillBlocked! Blocked (never moves)


!"

"

"!

!StillBlocked! Blocked (never moves)

!"

"

"! !

!So 2 blocking paths...

Blocking paths (both types) for one car when p = 1

For p close to 1, some will survive

Proof by comparison with oriented percolation on renormalized lattice...

Similar results for:

•Higher dimensions: jammed for p > p1(d)

•Biased initial conditions:

! w.p. p

" w.p. (1-) p

0 w.p. 1-p: jammed for p > p1()

•n-by-cn torus: jammed w.h.p. as n ! 1

•Random (Poisson) moves: jammed p > p1

Possible first step (question by B. Bollobas (?)): does every configuration in an n by n square (with free boundary conditions) clear within time 100n ?

seemingly yes for obvious “simple” configurations perhaps no (n log n ?) for random configurations!

(observations joint with O. Angel)

Prove anything for small p !

–asymptotically free-flowing ?

–not jammed ???

Open Problems

L by L square

L by L square

Cells initially occupiedindependently with probability p

L by L square


Dynamics: occ. ! occ.unocc. ! occ. if ¸2 occ. neighbours(update simultaneously

in discrete time)

X

L by L square

XDynamics: occ. ! occ.

unocc. ! occ. if ¸2 occ. neighbours(update simultaneously

in discrete time)


L by L square



in discrete time)


L by L square



in discrete time)


L by L square



in discrete time)


L by L square



in discrete time)


L by L square



in discrete time)


L by L square



in discrete time)

I(L,p) := P(entire square eventually occupied)


Simulations (Puzzle: minimum # of initiallyoccupied cells for entire squareto become occupied?)

I(L,p) := P(entire square eventually occupied)

L fixed, p ! 1 ) I ! 1L fixed, p ! 0 ) I ! 0

p > 0 fixed, L ! 1 ) I ! 1 (Van Enter 1987)

Proof: Z2, p > 0: show P(Z2 eventually occupied) = 1

>> 1/p

(L,p) ! (1,0):p log L > C1 ) I ! 1

p log L < C2) I ! 0

Theorem (Aizenman & Lebowitz 1988)

Question: can C1,C2 be replaced with C?

Yes:

Theorem (Holroyd 2003). (L,p) ! (1,0), > 0:

p log L > + ) I ! 1p log L < - ) I ! 0

where = 2/18.

(phasetransition)

(critical value)

Bootstrap percolationDiscrete time cellular automatonRandom starting state, deterministic evolutionMonotone, attractive, update order doesn’t matter

MotivationPhysical applications: magnetic alloys, storage arrays, hydrogen mixtures, glass transition, crack formation...Tool for analysis of other models: zero-temperature Ising model, diffusion models, kinetically constrained models...Step towards understanding other cellular automata...







Theorem: (L,p) ! (1,0):

p log L > + ) I ! 1p log L < - ) I ! 0

= 2/18.

Simulations (e.g. Adler, Stauffer, Aharony 1989)suggested

= 0.245 § 0.015

But

2/18 = 0.548311... !

Recall:

1 / log L

p1/2: I(L,p1/2)=1/2

Slope0.245

Slope2/18

L=28000L=1020

“crossover?”

Adler, Stauffer, Aharony 1989

Modified model:unocc. ! occ. if ¸ 1 occ. neighbour in each dimension

Theorem (Holroyd 2003):

p log L > + ) I ! 1p log L < - ) I ! 0

where = 2/6 = 1.644934...

X

Simulations:

= 0.47 § 0.02

Try to understand the discrepancy...

Other models (above: Z2, nearest neighbours, threshold =2)

Z2, n.n., =1:

Z2, n.n., =3:

grows forever

cannot be invaded

Zd, n.n., =d¸3: p log log L> C1 ) I! 1

< C2 ) I! 0

(L,p) ! (1,0)

(Cerf & Cirillo 1999, Cerf & Manzo 2002)d-1

Z2, asymmetric rules... p(log p) log L (Gravner, Griffeath, Mountford, Van Enter, Hulshof,...)

{0,1}n, n.n., = 2: p n2 4p n (Balogh &Bollobas 2003)

Conjecture: can replace C1,C2 by § in all above cases known only for:

X

1. Modified model in Zd (Holroyd 2006):p log log L = 2/6

d-1

2. Z2, , =k X

k-1(Holroyd, Liggett, Romik 2004):

p log L = 2/[3k(k+1)]

X3. Z2, , =4 (Duminil, Holroyd, in prep.):

p log L some

(Standard model in d¸3 is open,but 2/18, if it exists! – Holroyd, in prep.)

(Related to partition identities- Holroyd, 2008)

(Involves “irregular growth”)

Slow convergence

Z2, (L,p)! (1,0), p log L > + ) I! 1

Slow convergence

Z2, (L,p)! (1,0), p log L > ) I! 1

Slow convergence

Z2, (L,p)! (1,0), p log L > – c/plog L ) I! 1

Theorem (Gravner, Holroyd 2007) For standard or modified model:

[ - c’p p ]

Theorem (Gravner, Holroyd 2007) For modified model,version with (reasonable) explicit constants...

E.g. p1/2 log L < 0.99 for L = 103000 !!

(to halve “error” need L L4 !)(conjecture: 9 matching bound)

1/plog 28000 = 0.31...1/plog 1020 = 0.15...

Simulation/numerical estimates: standard: p1/2 log L ¼ 2/18 – 0.45/(log L)0.2 (Stauffer 2003) modified: p1/2 log L ¼ 2/6 – 3.67/(log L)0.333

(De Gregorio, Lawlor, Dawson 2006)

Theorem: modified model:

p log L > + ) I ! 1p log L < - ) I ! 0

where = 2/6.

Proof of:

i.e. 10

p! 0

main step:

Proposition: 8 (small) , 9 (large) B s.t. as p ! 0,

“density of nucleation centres”

Proposition ) Theorem:

L = exp(a/p)B/p

fills up <¼> contains a that fills itself up(nucleation centre)

E(# of nucleation centres) ¼ L2 I(B/p, p) ¼ exp(2a/p)

exp(-2/p) >> 1 if a >

<< 1 if a <

B large, p ! 0

L

B/p

Proof of:Proposition: 8 (small) , 9 (large) B s.t. as p ! 0,

lower bound:

I(B/p,p) ¸ P( ) = p 11 [1-(1-p)k]2

-p log P(...) ¼ - 11 2 log [1 - e-kp] p

2/6

Action happens at scales [c/p,C/p]

Upper bound:

Split event { fills up} into “ways” it could happen

Bound: # of waysprobability of each way

multiply

Hard part: find correct definition of “ways”...

B/p

Ideas: 1.Variational principles

P( ) ¼ P( ) >> P( ) , P( )

2. Coarse graining

¼

3. Necessary condition

fills up

fillsup

fillsup

disjointly

)

(L,p)! (1,0) , p log L > – c/plog L ) I! 1

Theorem for modified model:

Proof of:

Look for other growth mechanisms:

(rp)2

r

Prob. ¼

r << 1/p

p

Greater if r < p-1/2

If r < p-1/2/2, ratio

So,

p-1/2/2

B/p

P( )

More care gives better constant, explicit bounds.

The standard model.

Diagonal growth ) no saving. Previous proof does not work!

(rp)2 p2>

First , why 2/18 ?

no 2 adjacentcolumns vacantno 2 adjacent

columns vacant

-log P( ) » 2/[18p] p ! 0

= 2/18

(related to partition identities; Holroyd, Liggett, Romik 2004; Holroyd 2008)

k

3k(k+1)

Solution: another mechanism – the wiggle

d¼ 1/p

Cost:

> C ’’p if d < p-1/2

P( )/P( )

1/p

2/p

m = p-1/2/100 wiggles of scale < p-1/2

Choice of locations, sizes

Energy cost Entropy gain

I

pI(L,p) =

p1- log L – p log L = O(log log L / log L) (Balogh, Bollobas)

p log L1- – p log L = (1 / log L) (Gravner, Holroyd)

system size

para

mete

r

1-

Many models(e.g. Erdos-Renyi random graph)

window >> distance to limit

- Bootstrap percolation- Integer partitioning (Borgs, Chayes, Pittel)

window << distance to limit1/log L 1/plog L

p log L

2/18

Scaling window

L

Open problems:

Lower bound: p1/2 log L - ~ ? (Gravner, Holroyd, in prep: close bounds for “growth model”)

Tight bounds on p-window

Other models (d ¸ 3, ...)

Crossover – is there a “phantom slope” - prove in limit for a sequence of models?

Documents

Astonishing Cellular Automata Alexander E. Holroyd UBC and Microsoft