Deterministic Random Walks
Joshua Cooper
Benjamin Doerr
Joel Spencer
Gábor Tardos
UCSD (SC soon!)
MPI Saarbrücken
Courant Institute
Simon Fraser
An observation about cellular automata (see Wolfram’s NKS):
They generally fall into three categories.
t =1
t =2
t =3
An observation about cellular automata (see Wolfram’s NKS):
They generally fall into three categories.
I. Behavior so simple
we can prove that a
pattern emerges…
II. Behavior so complicated
you could simulate a Turing
machine on it…
III. And…
III. Behavior that is “randomlike”…
Such automata are useful:
1. Fast pseudorandom number generation
2. Quasi-Monte Carlo integration
3. Bounds in discrepancy theory / quasirandomness
However, very little is usually known outside of experimental results…
“The P-Machine”
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
10
1
9
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
1
9
1
8
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
1
8
1
7
1
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
1
7
1
6
1
1
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
1
1
6
2
5
1
1
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
2
1
5
2
4
1
1
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
2
1
4
2
3
2
1
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
2
1
3
2
2
2
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
2
2
2
3
1
2
2
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
3
2
1
3
2
2
t=0
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
3
2
3
2
t=1
1. At every step of (discrete) time, every chip moves.
2. When a single chip moves, it goes in the direction that
its “rotor” is pointing.
3. When a chip moves, its rotor turns 90°.
+.5
2.5
2.5
10
2.5
-.5 2.5
+.5
-.5
Compare to the “linear machine” : splits chips
evenly among neighbors.
How large can the difference be?
Same as the expected value for a simple random
walk on the graph.
Theorem 1 (C., Spencer ’05). The difference at any point, after any amount of
✴
time, with any initial configuration of chips, any initial configuration of rotors,
and any rotor permutations, is bounded by a constant cd that depends only on
the dimension d.
✴any even configuration.
Remark. This is best possible in the senses that:
a.) The statement is false for mixed even/odd configurations.
b.) cd is a computable constant, with c1 ≈ 2.29.
c.) The rotors can each go through a different permutation
of the 2d directions.
Amazingly, we can say something much stronger…
Restrict our attention to
d = 1, i.e., a P-machine on the integers:
Definition. Write Δ(x,t) for the discrepancy between the P-machine and the linear
machine at the point x at time t.
Definition. Write Δ(S,Z) for the discrepancy on a set S over all times in
( S , Z )   ( x, t )
xS tZ
Z, i.e.,
Theorem (C., Doerr, Tardos, Spencer) :
L∞ for Space-Intervals
( I , t )  O(log L)
for intervals I of length L.
Theorem (CDTS) :
L2 for Space-Intervals
1
M
M
2

(
I

x
,
t
)
 O(log L)

x 1
for intervals I of length L, and
M sufficiently large.
Corollary (CDTS) : For “most” translates of an interval,

( I , t )  O log L

Theorem (CDTS) :
L∞ for Time-Intervals
 ( x, J )  O ( T )
for intervals J of length T.
Theorem (CDTS) :
L∞ for Space-Time-Intervals

 cT log L / T
( I , J )  
c L T

if L  e T
if L  e T
for intervals I of length L and intervals J of length T.
Not only that… but ALL of these results are best possible.
That is, there exist (different) initial configurations of chips and rotors so that, for
any given intervals I, J with lengths L and T, respectively,
( I , t )  (log L)
1
M
M
2

(
I

x
,
t
)
 (log L)

x 1
 ( x, J )   ( T )

 cT log L / T
( I , J )  
c L T

if L  e T
if L  e T
The upper bounds are proved by counting the contributions
to the final quantity that each chip makes at each time.
Lots of cancellation translates to small discrepancies.
For the lower bounds, we show that all the arguments can
be reversed, i.e., there is a sequence of chips-and-arrows
so that the upper bound is achieved.
Two crucial tools...
Theorem (CDST) : Parity-Forcing
: ℤ × ℕ0 → {0, 1}, there
exists an initial even configuration of the chips such that for all x ℤ,
t ℕ0 such that x ≡t (mod 2), we have chips(x, t) ≡ (x, t) (mod 2).
For any initial position of the arrows and any
This follows from the following statement…
Theorem (CDST) : Arrow-Forcing
Let ρ: ℤ × ℕ0 → {left,
right} be defined arbitrarily. There exists an even
initial configuration that results in the arrows agreeing with ρ(x, t) for all x and t
with x ≡t (mod 2).
The proof would have been easier if only…
Definition: p(v, t) is the probability that a chip leaving from 0 arrives at
v at time t in a simple random walk
Conjecture: The probability that v is visited at time t in a random walk
started from the origin, p(v,
For a function χ : ℤd
t), is unimodal (in t 2ℤ).
→ ℝ, define
p(  , t ) 
  ( v ) p ( v, t )
vZ d
p(χ, t) is the concatenation of a finite number of
monotone subsequences, depending only on |supp(χ)|.
Conjecture:
This set-up can be vastly generalized:
Given a graph G, and functions
f : V(G) → ℕ0
r : V(G) → V(G)*
with r(v) a permutation of N(v)
the initial number of chips
the rotor sequences
Define chips(x,t) = chip count at x at time t for a P-machine on G.
Define E(x,t) = chip count at x at time t for a linear machine on G.
Question: For which bipartite G must chips(x,t) - E(x,t) remain bounded for
any x, t, r, and f with supp(
f ) in one color class?
Theorem (CDS’05): Not the infinite regular tree.
Wild and Unfounded Guess: It has something to do with amenability.
THANK YOU!