Occupation probabilities for
coalescing (annihilating) Brownian
motions on R
Roger Tribe and Oleg Zaboronski
Department of Mathematics, University of Warwick
Warwick, 19.03.2012 – p. 1/14
Plan
Quantities
Coalescing (annihilating) Brownian motions (CABM)
Review of known results
The main result for occupation probabilities
Structures
Pfaffian representation for occupation probabilities
Duality and the maximal entrance law
Pfaffian point processes and CABM
Conclusions
Warwick, 19.03.2012 – p. 2/14
CABM’s
Particles perform
independent Brownian
motions on R until they
meet
At the moment of collision
particles instantly
coagulate. The aggregate
follows a Brownian path
with the same diffusion
rate
Thinning relation:
D
A
under PΘ(Ω) =
Θ(Xti ) underPΩC
(Xti )
Proof :Colouring
The main object of interest:
ρn (t, x1 , . . . , xn )dx1 . . . dxn the probability of finding n
particles in dx1 , . . . , dxn at
time t
Warwick, 19.03.2012 – p. 3/14
Mean field analysis
’Chemist’: Aggregation rate is proportional to the
number of pairs of particles within the interaction range
Hence the rate equation: ∂t ρ1 (t) = −λ(r, D)ρ1 (t)2
Solution: ρ1 (t) ∼
1
t
Implicit mean field assumption: ρn (t) ∼ ρ1 (t)n
Consequently, ρn ∼
1
tn
Warwick, 19.03.2012 – p. 4/14
The breakdown of mean field theory: anti-correlations
Mean field
Spatially extended
1.4
1.2
PM(x)/ <PM>
1
0.8
0.6
0.4
0.2
0
10
20
30
40
50
60
70
80
90 100
x/ ∆ x
(Monte-Carlo simulations on Z. Courtesy of Colm
Connaughton)
Warwick, 19.03.2012 – p. 5/14
Known rigorous results for t → ∞
d=1
ρ1 (t) ∼
1
√
t
(Bramson-Griffeath,
Z, 1980;
Bramson-Lebowitz,
1988)
Exact expression for
ρn (t) for all t’s using
empty interval method
in the form of the sum
of products of duffusive Greens’ functions
with alternating signs.
(ben-Avraham, 1998)
d≥2
For d = 2, ρ1 (t) ∼ ln(t)
t
(Bramson-Lebowitz,
1988)
For d > 2, ρ1 (t) ∼
1
t (Bramson-Lebowitz,
1988; van den Berg
and Kesten, finite reaction rates, 2002)
Warwick, 19.03.2012 – p. 6/14
Predictions from (non-rigorous) renormalization group analysis

n(n−1)
n

−2− 4

d
=
1
t


n(n−1)
n
ln t
− 2
ρn(t) ∼
(ln
t)
d
=
2
t



 t−n
d>2
Note the non-linear dependence of the scaling
exponent on n in d = 1
Warwick, 19.03.2012 – p. 7/14
Multi-scaling of occupation probabilities
Theorem 1 Under the maximal entrance law for C(A)BM’s,
(2n)
x
−n
sup ρt (x1 , x2 , . . . , x2n ) − cn t ∆2n √ → 0 as t → ∞,
t
|xi |<<t1/2
∆2n (x) =
Q
ABM =
(x
−
x
)
,
c
i
j
n
1≤i<j≤2n
1 CBM
4 n cn
IC’s: Maximal entrance law initial conditions (Arratia,
1981): ’one particle per site’ at t = 0 - as in Brownian
web
Construction: Poisson(λ) initial distribution with λ → ∞
Warwick, 19.03.2012 – p. 8/14
Pfaffians and interacting particle systems
Theorem 2 Consider a system of ABM’s with 2n particles
at t = 0, x1 < x2 < . . . x2n . Then the product moment
Q
(2n)
A
i
mt (x1 , . . . , x2n ) = E(x1 ,...,xn )
i∈It g(Xt ) is given by the
Pfaffian of an 2n × 2n antisymmetric matrix:
(2n)
(2)
mt (x1 , . . . , x2n ) = P f (−1)χ(j>i) mt (xi , xj )
(2n)
Proof. mt
(x1 , . . . , x2n ) solves heat equation on the cell
(2n)
(2n−2)
x1 < x2 < . . . x2n ⊂ R2n . BC’s: mt |xi =xi+1 = mt
. IC’s:
Q2n
(2n)
m0 = k=1 g(xk ). Pfaffian solves the equation and
satisfies IC’s, BC’s. The theorem follows by uniqueness
Warwick, 19.03.2012 – p. 9/14
Coalescing Brownian motions and Pfaffians
CBM-ABM duality (Arratia) for the maximal entrance
law:
C
A
P∞
[Nt [a1 , a2 ] = 0 . . . Nt [a2n−1 , a2n ] = 0] = P(a
(τ < t)
i)
The right hand side is the Pfaffian of
(Brownian hitting prob)
Proof: set g ≡ 0 in Thm 2
(A)
Pai ,aj (τ
< t)
Conclusion: empty interval probabilities are Pfaffians
Warwick, 19.03.2012 – p. 10/14
CBM’s and Pfaffian point processes.
Theorem 3 Under the maximal entrance law for coalescing
Brownian motions, the particle positions at time t form a
Pfaffian point process with kernel t−1/2 K(xt−1/2 , yt−1/2 ),
where
!
−F ′′ (y − x)
−F ′ (y − x)
K(x, y) =
F ′ (y − x) sgn(y − x)F (|y − x|)
π −1/2
R∞
2
−z
e /4 dz .
and F (x) =
(Here sgn(z) = 1 for z > 0,
x
sgn(z) = −1 for z < 0 and sgn(0) = 0.)
Proof: Differentiate the pfaffian expression for empty interval
probabilities with respect to right end points.
Closing the loop: Theorem 1 follows from the large-t
expansion of the pfaffian formulae for ρn ’s
Warwick, 19.03.2012 – p. 11/14
CABM’s and random matrices
Corollary 4
KtABM (x, y)
1
Ginibre
= √ Krr
2t
x
y
√ ,√
2t 2t
,
Ginibre is the N → ∞ limit of the Kernel of the
where Krr
Pfaffian point process characterising the law of real
eigenvalues in the real Ginibre(N ) ensemble,
1
− 21 T r(MT M)
µN (dM) =
λN ×N (dM)
2 /2 e
N
(2π)
(Ginibre, Edelman, Sommers, Akemann, Forrester, Sinclair,
Borodin, ...)
Warwick, 19.03.2012 – p. 12/14
Conclusions
Mean field approximation in d = 1 is invalidated by
strong negative correlations between the particles
Multi-point probability densities exhibit quadratic
multi-scaling
One-dimensional occupation densities in CBM’s are a
Pfaffian point process
The same process describes occupation densities of
real eigenvalues in N → ∞ limit of real Ginibre matrix
ensemble
Warwick, 19.03.2012 – p. 13/14
Open questions, references
Is there a relation between CABM’s and the
GL(N )-valued Brownian Motions? ("Ginibre process").
Conjecture presented in [2] incorrect
Rigorous derivation of logarithmic corrections in d = 2?
References:
1. Multi-Scaling of the n-Point Density Function for
Coalescing Brownian Motions , CMP Vol. 268, No. 3,
December 2006;
2. Pfaffian formulae for one dimensional coalescing
and annihilating systems, arXiv Math.PR:
1009.4565; EJP, vol. 16, Article 76 (2011)
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