Outline
Spin variables
Interacting particle systems and Pfaffians
Annihilating and coalescing particle systems as
extended Pfaffian point processes
Roger Tribe, Kwan Yip and Oleg Zaboronski
Department of Mathematics, University of Warwick
May 6 2013
Venice, May 6-10, 2013
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Outline
1
Spin variables
Definition
From spin variables to correlation functions
2
Interacting particle systems and Pfaffians
Pfaffians
Annihilating Brownian motions on R
Pfaffian Point Processes
Extended Pfaffian Point Process
Applications
3
Conclusions
Venice, May 6-10, 2013
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Spin variables
N(dx)
simple point measure on R
Spin variable s(x) = (−1)N(0,x)
x <y
implies s(x)s(y ) = (−1)N(x,y )
Spin correlations
Venice, May 6-10, 2013
E [s(x1 ) . . . s(x2n )]
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Spin variables → correlation functions
d
(−1)N(x,y ) = (−2)(−1)N(x,y ) N(dx)
dx
1
d
− lim s(x)s(y ) = N(dx)
2 y ↓x dx
ρ(x1 , x2 , . . . , xn )
n
1
=
−
lim . . . lim
x2n ↓x2n−1
2 x2 ↓x1
d d
d
...
E [s(x1 ) . . . s(x2n )]
dx1 dx3
dx2n−1
Venice, May 6-10, 2013
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Pfaffians
Recall, for anti-symmetric real 2n × 2n matrix A,
det(A) = (Pf(A))2 .

0
a
 −a 0
Pf 
 −b −d
−c −e
Venice, May 6-10, 2013
b
d
0
−f

c
e 
 = af − be + cd.
f 
0
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Annihilating Brownian motions on R
Nt (A) = Number of particles in A at time t
E[s(x1 ) . . . s(x2n )] = Pf (E[s(xi )s(xj )] : i < j)
Proof: Both sides solve
1
∂t u (2n) = ∆u (2n)
2
with boundary conditions
on V2n = {x1 < x2 < . . . x2n }
u (2n) |xi =xi +1 = u (2n−2) (t, x1 , . . . , xi −1 , xi +2 , . . . , x2n )
Venice, May 6-10, 2013
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Conclusions
Pfaffian Point Processes
.
Corollary: Nt is a Pfaffian Point process.
.
That is, ρt (x1 , x2 , . . . , xn ) = Pf(K (xi , xj ) : i < j)
Special case: maximal entrance law
Kt (z) =
Kt11 (z) Kt12 (z)
Kt21 (z) Kt22 (z)
=
−F
′′ (zt −1/2 )
t −1
F ′ (zt −1/2 )
t −1/2
and F is the Gaussian error function given by
Z ∞
1
2
F (z) = 1/2
e −x /4 dx
2π
z
Venice, May 6-10, 2013
−F
′ (zt −1/2 )
t −1/2
sgn(z)F (|z|t −1/2 )
!
Outline
Spin variables
Interacting particle systems and Pfaffians
Extended Pfaffian Point process
Let
ρ((t1 , x1 ), . . . , (tn , xn ))dx1 . . . dxn
= P(particles at times ti at positions dxi )
Then
ρ((t1 , x1 ), . . . , (tn , xn )) = Pf(K ((ti , xi ), (tj , xj ) : i < j))
Under maximal entrance law: for t > s and i , j ∈ {1, 2}
K ij ((t, x); (s, y )) = Gt−s Ksij (y − x) − 2I{i =1,j=2} gt−s (y − x);
Proof: double induction over space and time points.
Venice, May 6-10, 2013
Conclusions
Outline
Spin variables
Interacting particle systems and Pfaffians
Conclusions
Applications
Coalescing case: The same structure as ABM’s. The kernel
is rescaled by 2.
Proof. Use empty interval formula
I (Nt ([y1 , y2 ]) = Nt ([y3 , y4 ]) = . . . = Nt ([y2m−1 , y2m ]) = 0)
in place of product spin formula.
Negative dependence:
An x ρt (x1 , x2 , . . . , xn ) = n/2 ∆ √ 1 + O(t −1/2 )
t
t
Venice, May 6-10, 2013
Outline
Spin variables
Interacting particle systems and Pfaffians
Conclusions
Conclusions
Coalescing (annihilating) Brownian motions on R can be
characterized as an extended Pfaffian point process
The one dimensional law of C(A)BM’s at t = 1 coincides with
the law of real eigenvalues for the real Ginibre ensemble in the
limit N → ∞ (Borodin, Sinclair; Forrester, Nagao)
The Pfaffian structure allows for a detailed study of the
structure of correlations in C(A)BM’s including negative
dependencies between particles
Further research: asymptotics of multi-time correlations, the
distribution of inter-particle spacings (Janossi densities),
models with immigration
Reference: Roger Tribe, Oleg Zaboronski, K. Yip, One
dimensional annihilating and coalescing particle systems as
extended Pfaffian point processes, ECP vol.17 (2012)
Venice, May 6-10, 2013