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Probabilistic Proofs of Combinatorial Identities*
Jessica Jay, University of Bristol
Probability, Analysis and Dynamics Conference 2022
*Based on joint work with M ́arton Bal ́azs and Dan Fretwell, arXiv:2011.05006 (2021)
Probabilistic Proofs of Combinatorial Identities*
Jessica Jay, University of Bristol
Probability, Analysis and Dynamics Conference 2022
*Based on joint work with M ́arton Bal ́azs and Dan Fretwell, arXiv:2011.05006 (2021)
Asymmetric Simple Exclusion
(ASEP)
Nearest neighbour interacting particle system on Z with at
most one particle per site:
LMP RMH Z
q
−1
↷
q
↶
0 < q < 1
↷X
Product Bernoulli stationary blocking measure:
μ
c
(η) = Y
∞
i=−∞
q
−2(i−c)ηi
1 + q
−2(i−c)
.
Conserved Quantity: N(η) = P
∞
i=1
(1 − ηi) −
P
0
i=−∞
ηi
Ergodic Decomposition of state space:
Ω = [
n∈Z
Ω
n where Ωn
:= {η ∈ Ω : N(η) = n}.
Conditional stationary measure:
ν
n
(η) = μ
c
(η|N(η) = n) =
P
∞
m=−∞
q
m(m+1)−2mc
q
n(n+1)−2nc
· μ
c
(η).
Asymmetric Zero-Range (AZRP)
Nearest neighbour interacting particle system on Z<0 with
any number of particles per site and open right boundary:
LMP −1 0
q
−1
↷
q
↶
Product Geometric stationary blocking measure:
π(ω) =
Y
∞
i=1
q
2ω−i
(1 − q
2i
).
ASEP-AZRP Correspondence
Exclusion and Zero-range are in correspondence (when
restricting to a given conserved quantity).
The number of holes between particle i and i + 1 in ASEP is
the number of particles at the site −i in AZRP
Jacobi Triple Product
Theorem.
X
m∈Z
q
m2
z
m
=
Y
∞
i=1
(1 − q
2i
)(1 + q
2i−1
z)(1 + q
2i−1
z
−1
)
We can prove this by considering the ground states
of ASEP and AZRP
−5 −4 −3 −2 −1 0 1 2 3 4 5
Ground state η
0 ∈ Ω
0
of ASEP
−10−9 −8 −7 −6 −5 −4 −3 −2 −1 0
Ground state ω
0
of AZRP
Correspondence: ν
0
(η
0
) = π(ω
0
), explicitly:
P
m∈Z
q
m2+(1−2c)m
Q
(1 + q
2i−1+(1−2c)
)(1 + q
2i−1−(1−2c)
)
=
Y
∞
i=1
(1−q
2i
).
Letting z := q
1−2c
and rearranging completes the
proof. (Bal ́azs and Bowen 2018) □
Family of 0-1-2 Systems
Nearest neighbour interacting particle systems on Z with up to 2 particles per site:
LMP RMH Z
p(2,0)
q(0,1) ↷
↶
↷X
Assumptions on rates:
•Asymmetric: the system has right drift
•Attractive: p(2, ·) ≥ p(1, ·) > 0 and p(·, 0) ≥ p(·, 1) > 0
q(0, ·) ≥ q(1, ·) > 0 and q(·, 2) ≥ q(·, 1) > 0
•Algebraic Conditions: p(1,0)
q(0,1) =
p(2,1)
q(1,2) and p(1,0)p(2,1)q(1,1)q(0,2)
q(0,1)q(1,2)p(2,0)p(1,1) = 1
Product Stationary Blocking Measure:
μ
c
(η) = Y
∞
i=−∞
t
I{ηi=1}
q ̃
−(i−c)ηi
1 + tq ̃
−(i−c) + q ̃
−2(i−c)
Conserved Quantity: N(η) = P
∞
i=1
(2 − ηi) −
P
0
i=−∞
ηi
Conditional stationary measure (n = 0):
ν
0
(η) =
2
P
∞
l=−∞
q ̃
l(l+1)−2lcμ
c
(η)
1 + Q
∞
i=−∞
(1 − 2μ
c
i
(1))
Corresponding Family
By considering the inter-particle distances in the original process we find a family of
restricted nearest neighbour interacting particle systems on Z<0 with any number of
particles per site and open right boundary:
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
Restriction: No two consecutive empty sites!
Dynamics: The dynamics are inherited from the original process i.e. the jump rates can
be written down in terms of the original jump rate functions p(·, ·) and q(·, ·).
Product Stationary Measure:
π
e
(ω) =
q ̃
P
i odd
iω−i+
P
i even
i(ω−i−1)
t
2
P
i odd
I{ω−i≥1}−
P
i even
I{ω−i=0}
Seven(q, t ̃ )
Here we have focused on the states that come from standing up states with even conserved
quantity. We can do a similar thing with odd conserved quantity; this just shifts the circled
pattern above onto the odd sites.
A New Identity!
Ground states:
−5 −4 −3 −2 −1 0 1 2 3 4 5
η
e
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1
ω
e
Using the correspondence we have that
ν
0
(η
e
) = π
e
(ω
e
),
rearranging and letting z := q ̃
−c we have proved:
2
X
l∈Z
Seven(q, t ̃ )q ̃
l(l+1)z
2l =
Y
i≥1
(1 + tzq ̃
i + z
2
q ̃
2i
)(1 + tz−1
q ̃
i−1 + z
−2
q ̃
2(i−1))
+
Y
i≥1
(1 − tzq ̃
i + z
2
q ̃
2i
)(1 − tz−1
q ̃
i−1 + z
−2
q ̃
2(i−1))
Generalized Frobenius Partitions (GFP’s)
Frobenius partitions, of n, are two row arrays which correspond to ordinary partitions of n (see left picture below); these arrays
have strictly decreasing rows ai and bi and are such that n = s +
P
s
i=1
(ai + bi). The Jacobi Triple Product Identity gives an
equivalence of generating functions for ordinary partitions.
Partition of 30
Corresponding GFP of 30
7 6 4
6 3 1
4 2 2
2 2 0
Corresponding generalised
Young diagram
GFP (with repeats)
of 15
We can generalise the notion of Frobenius partition to two-rowed arrays with certain conditions on the rows. For example, we
take ai and bi to be non-increasing such that no part repeats more than once in a row. These no longer correspond to ordinary
partitions but can be represented by some generalised Young diagram (see right picture above). The identity found by considering
the 0-1-2 family gives an equivalence of generating functions for these GFP’s (the power of t gives the number of distinct parts).
t :=
p(1, 0)q(0, 2)
q(0, 1)p(1, 1)
1
2
≥ 1
q ̃ :=
q(0, 1)
p(1, 0) < 1
We must consider the cases
when n ∈ 2Z and n ∈ 2Z + 1
separately.
We always see this pattern far to the left.
Makes finding stationary distribution more complicated.
A similar identity is proved by considering the odd ground states. These
identities are new and specialise to give known 2-variable identities.