Joshua Cooper Benjamin Doerr Joel Spencer Gábor Tardos Deterministic Random Walks UCSD (SC soon!) MPI Saarbrücken Courant Institute Simon Fraser.

Joshua CooperBenjamin DoerrJoel SpencerGábor Tardos

Deterministic Random Walks

UCSD (SC soon!)MPI SaarbrückenCourant 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 simplewe can prove that apattern emerges…

II. Behavior so complicatedyou could simulate a Turingmachine 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°.

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°.

9110

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°.

819

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°.

718

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

1617

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

1

1 526

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

1

1 425

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

1

1 324

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

1 223

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

2 132

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

2

2 31

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

3

2

2

t=1

Compare to the “linear machine” : splits chipsevenly among neighbors.

2.5

2.5

2.5

2.5

Same as the expected value for a simple randomwalk on the graph.

10 +.5

-.5

-.5

+.5

How large can the difference be?

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.

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

✴

✴any even configuration.

the dimension d.

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 Z, i.e.,

Sx Zt

txZS ),(),(

Theorem (C., Doerr, Tardos, Spencer) : L∞ for Space-Intervals

)(log),( LOtI

for intervals I of length L.

Theorem (CDTS) : L2 for Space-Intervals

)(log),(1

1

2 LOtxIM

M

x

for intervals I of length L, and M sufficiently large.

Corollary (CDTS) : For “most” translates of an interval,

LOtI log),(

Theorem (CDTS) : L∞ for Space-Time-Intervals

TeLTLc

TeLTLTcJI

if

if /log),(

for intervals I of length L and intervals J of length T.

Theorem (CDTS) : L∞ for Time-Intervals

)(),( TOJx

for 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,

)(log),( LtI

)(log),(1

1

2 LtxIM

M

x

)(),( TJx

TeLTLc

TeLTLTcJI

if

if /log),(

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

For any initial position of the arrows and any : 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).

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 twith x ≡ t (mod 2).

This follows from the following statement…

The proof would have been easier if only…

For a function χ : ℤd → ℝ, define

dv

tvpvtpZ

),()(),(

Conjecture: p(χ, t) is the concatenation of a finite number of

monotone subsequences, depending only on |supp(χ)|.

Conjecture: The probability that v is visited at time t in a random walk

started from the origin, p(v, t), is unimodal (in t 2ℤ).

Definition: p(v, t) is the probability that a chip leaving from 0 arrives at

v at time t in a simple random walk

This set-up can be vastly generalized:

Given a graph G, and functions

f : V(G) → ℕ0 the initial number of chips

r : V(G) → V(G)* with r(v) a permutation of N(v) 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?

Wild and Unfounded Guess: It has something to do with amenability.

Theorem (CDS’05): Not the infinite regular tree.

THANK YOU!