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)
Another way for a car to be blocked:
!"
"
"! !
!You Blocked (never moves)
If this never moves,you are blocked
Another way for a car to be blocked:
!"
"
"! !
!You Blocked (never moves)
If this does move....
Another way for a car to be blocked:
!"
"
"! !
!You Blocked (never moves)
Another way for a car to be blocked:
!"
"
"! !
!You Blocked (never moves)
Another way for a car to be blocked:
!"
"
"!
!StillBlocked! Blocked (never moves)
Another way for a car to be blocked:
!"
"
"!
!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
Cells initially occupiedindependently with probability p
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)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
Cells initially occupiedindependently with probability p
L by L square
XDynamics: occ. ! occ.
unocc. ! occ. if ¸2 occ. neighbours(update simultaneously
in discrete time)
I(L,p) := P(entire square eventually occupied)
Cells initially occupiedindependently with probability p
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...
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...
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...
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?
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