poisson matching alexander e. holroyd (ubc) joint with: robin pemantle, yuval peres, oded schramm

Poisson Matching

Alexander E. Holroyd (UBC)

Joint with:Robin Pemantle, Yuval Peres,Oded Schramm

Red points

Blue points

Perfect matching

How short can wemake the edges?

?

Random perfectmatching scheme M

Rate-1 Poisson processR of red points

Independentrate-1 Poisson processB of blue points

Assume (R, B, M)translation-invariantin law

Rd

Example: Gale-Shapley stable matching.

- Match all mutually closest red/blue pairs.



- Remove them

- Repeat indefinitely



- Remove them


Alternative description: ball-growing



- Remove them


Alternative description: ball-growing

Alternative description: unique matching with no unstable pairs

Two-colourstablematching

(on torus)

Two-colourminimum-lengthmatching

(on torus)

One-colourstablematching

(on torus)

One-colourminimum-lengthmatching

(on torus)

Call a matching scheme - a factor if M = f(R, B) (e.g. stable matching)

- randomized if not

Given a matching scheme M,

denote X = length of “typical edge”

i.e. P*(X · r) := E # {red points z 2 [0,1)d with |z-M(z)| ·

r}

Main question: how small can we make X (in terms of tail behaviour)?

A trivial lower bound: for any matching,P*(X > r) ¸ P*(9 no other point in B(0,r)) ¸ e-

crd

i.e. E* ecXd = 1

= |0-M(0)| “conditioned” on {0 is red} (Palm measure P*)

0

X

One color Lower bound Upper boundRandomized

d=1d¸2

Factor d=1d¸2

Stable All d

Two color Lower bound Upper boundRandomized

d=1d=2d¸3

Factor d=1d=2d¸3

Stable d=1d=2d¸3


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1

Stable All d E* ecXd = 1


d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

Factor d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

Stable d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

P*(X > r) < C r-d/2

Factor d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

Stable d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

P*(X > r) < C r-d/2

Factor d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

Stable d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1

Stable d=1d=2d¸3

E* ecXd = 1

E* ecXd = 1

E* ecXd = 1


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-2/3+ [Soo]P*(X > r) < C r-2d/(d+4)+[Soo]

Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecXd = 1

E* ecXd = 1

Factor d=1d¸2

E* ecXd = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1 [Timar]

E* eCXd-2 < 1 [Timar]

Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

Factor d=1d¸2

E* ecX = 1

E* ecXd = 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1

Stable All d E* Xd = 1 P*(X > r) < C r-d


d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

Two-color randomized matching

12

4

Invariant randomdyadic partitioning

....

Match as much as possible within level-1 cubes

Match as much as possible within level-2 cubes

Etc.

P*(X>c2k) · P*(0 not matched by stage k) · P*(0 in “excess” in its level-k cube) · C p[2kd]/2kd = C(2k)d/2

Two-color randomized matching: better method for d¸3

12

4


....

(based on Ajtai-Komlos-Tusnady)


12

4


....

Repartition to equalize points per unit volume,affinely shift points



12

4


....


Iterate



12

4


....


Iterate...

Get allocation of 1 unit volume to each point... abstract arguments ) matching


Total distance moved by a typical point ¼

V § V1/2

Volume V

V1/d-1/2

< 1 for d ¸ 3

Biggest deviation: empty cubes...


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

Two-color lower bound, d=2

Directed linesegment u

K(u) := # edges intersecting u with red to the left

Assume E*X < 1. Then E K(u) < 1

Assume matching ergodic. Then

n

# - # = # - #

= 14 K(si) - 1

4 K(-si)

s1

s2

s3

s4

E( ) = (n)E( ) = o(n) #


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

1-color, 1 dimension

(Alternating matching)

O

1/2

1/2

) 9 a randomized matching with P*(X > r) = e-r

@ a factor alternating matching

Any factor matching has E*X = 1. Proof:


O

Enough to show:

E(# edges crossing O) = 1


O

Enough to show:

P(# edges crossing O = 1) = 1


O

Suppose:

P(# edges crossing O = k) > 0

k k k k

even #

Rematch ) factor alternating matching! #


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

Two-color stable matching – lower bound

Claim: E(# red points that prefer some part of B) = 1

B=B(0,1) Implies E*Xd=1


Prove: P(¸ k red points prefer some part of B) = 1





Add k extra blue points in BLaw abs. cts. wrt Poisson

new points all get matched in the stable matching

Fact: adding blue points makes red points happier




Add k extra blue points in BLaw abs. cts. wrt Poisson

new points all get matched in the stable matching

Fact: adding blue points makes red points happier

So k red partners preferred part of B before


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

Two-color stable matching – upper bound, d=1

Bad red point: has edge of length > r

0 r

I

Fact: edges cannot cross:

So other red and blue points in I equalize

r P*(X>r) = E(# bad points in [0,r]) = E(#red – #blue in I) · E max J½ [0,r] (#red - #blue in J) · range of RW

» Cpr


d=1d¸2

E* ecX = 1

E* ecXd = 1

E* eCX < 1

E* eCXd < 1

Factor d=1d¸2

E* X = 1

E* ecXd = 1

P*(X > r) < C r-1

E* eCXd < 1



d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-1

E* eCXd < 1

Factor d=1d=2d¸3

E* X1/2 = 1E* X = 1

E* ecXd = 1

P*(X > r) < C r-1/2



Stable d=1d=2d¸3

E* X1/2 = 1E* X = 1E* Xd = 1

P*(X > r) < C r-1/2

P*(X > r) < C r-0.496...

P*(X > r) < C r-s(d)

Two-color stable matching – upper bound, d=2

B=B(0,R)bad if it prefers all of B

Call a red or blue point in B

funny if good but partner is outside B

# bad red · # funny blue + (#red - #blue)+

E(# bad red) · E(# funny blue) + CpR r:=P*(X>r)

Iterate to get 2k · (2k)-s More care gives s=0.496...

poisson matching alexander e. holroyd (ubc) joint with: robin pemantle, yuval peres, oded schramm

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