Solution to a problem arising
from Mayer’s theory of cluster integrals
Olivier Bernardi, C.R.M. Barcelona
October 2006, 57th Seminaire Lotharingien de Combinatoire
CRM, Barcelona
Olivier Bernardi – p.1/37
Content of the talk
Mayer’s theory of cluster integrals
Pressure = Generating function of weighted connected
graphs.
CRM, Barcelona
OOlivier Bernardi – p.2/37
Content of the talk
Mayer’s theory of cluster integrals
Pressure = Generating function of weighted connected
graphs.
Hard-core continuum gas
Pressure = Generating function of Cayley trees.
CRM, Barcelona
OOlivier Bernardi – p.2/37
Content of the talk
Mayer’s theory of cluster integrals
Pressure = Generating function of weighted connected
graphs.
Hard-core continuum gas
Pressure = Generating function of Cayley trees.
Why ?
[Labelle, Leroux, Ducharme : SLC 54]
CRM, Barcelona
OOlivier Bernardi – p.2/37
Content of the talk
Mayer’s theory of cluster integrals
Pressure = Generating function of weighted connected
graphs.
Hard-core continuum gas
Pressure = Generating function of Cayley trees.
Why ?
[Labelle, Leroux, Ducharme : SLC 54]
Combinatorial explaination
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Olivier Bernardi – p.2/37
Mayer’s theory of cluster integrals
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Olivier Bernardi – p.3/37
Statistical physics
Gas of n particules in a box Ω.
x2
Ω
x1
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x3
OOlivier Bernardi – p.4/37
Statistical physics
Gas of n particules in a box Ω.
x2
Ω
x1
x3
The energy of a configuration x1 , . . . , xn is
X
X
(x1 , . . . , xn ) =
µ(xi ) +
φ(xi , xj ).
i
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i<j
OOlivier Bernardi – p.4/37
Statistical physics
Gas of n particules in a box Ω.
x2
Ω
x1
x3
The energy of a configuration x1 , . . . , xn is
X
X
(x1 , . . . , xn ) =
µ(xi ) +
φ(xi , xj ).
i
i<j
No external field : µ(xi ) = µ.
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Olivier Bernardi – p.4/37
Statistical physics
x2
Ω
x3
x1
Energy : (x1 , . . . , xn ) = nµ +
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P
i<j
φ(xi , xj ).
OOlivier Bernardi – p.5/37
Statistical physics
x2
Ω
x3
x1
Energy : (x1 , . . . , xn ) = nµ +
P
i<j
φ(xi , xj ).
The partition function is
ZZ
1
(x1 , . . . , xn )
Z(Ω, T, n) =
exp −
dx1 ..dxn .
n! Ωn
kT
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OOlivier Bernardi – p.5/37
Statistical physics
x2
Ω
x3
x1
Energy : (x1 , . . . , xn ) = nµ +
P
i<j
φ(xi , xj ).
The partition function is
ZZ
1
(x1 , . . . , xn )
Z(Ω, T, n) =
exp −
dx1 ..dxn
n! Ωn
kT
ZZ Y
1
φ(xi , xj )
= n
exp −
dx1 ..dxn .
λ n! Ωn i<j
kT
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Olivier Bernardi – p.5/37
Example
Hard particules in Ω = {1, . . . , q}.
λ = 1 and φ(x, y) = +∞ if x = y
0
otherwise.
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OOlivier Bernardi – p.6/37
Example
Hard particules in Ω = {1, . . . , q}.
λ = 1 and φ(x, y) = +∞ if x = y
0
otherwise.
The partition function : Y
X
1
φ(xi , xj )
Z(Ω, T ) ≡
exp −
n! Ωn i<j
kT
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OOlivier Bernardi – p.6/37
Example
Hard particules in Ω = {1, . . . , q}.
λ = 1 and φ(x, y) = +∞ if x = y
0
otherwise.
The partition function : Y
X
1
φ(xi , xj )
q
Z(Ω, T ) ≡
exp −
=
n! Ωn i<j
kT
n
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Olivier Bernardi – p.6/37
Mayer’s Idea (1940)
φ(xi , xj )
exp −
kT
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= 1 + f (xi , xj ).
OOlivier Bernardi – p.7/37
Mayer’s Idea (1940)
φ(xi , xj )
exp −
kT
i<j
Y
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=
Y
i<j
1+f (xi , xj ) =
X
Y
f (xi , xj ).
G⊆Kn (i,j)∈G
OOlivier Bernardi – p.7/37
Mayer’s Idea (1940)
φ(xi , xj )
exp −
kT
i<j
Y
=
Y
1+f (xi , xj ) =
i<j
X
Y
f (xi , xj ).
G⊆Kn (i,j)∈G
⇒ Partition function can be written as a sum over graphs :
ZZ Y
1
φ(xi , xj )
Z(Ω, T, n) ≡ n
exp −
dx1 ..dxn
λ n! Ωn i<j
kT
1 X
= n
W (G),
λ n! G⊆K
n
where W (G) =
ZZ
Y
f (xi , xj )dx1 ..dxn
Ωn (i,j)∈G
is the Mayer’s weight of G.
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Olivier Bernardi – p.7/37
For those familiar with the Tutte Polynomial
Mayer’s tranformation is the analogue (for general partition
function) of the correspondence
Partition function of the Potts model
(coloring expansion)
⇐⇒
Tutte polynomial
(subgraph expansion)
[Fortuin & Kasteleyn 72]
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Olivier Bernardi – p.8/37
Example
Hard particules in Ω = {1, . . . , q}.
φ(x, y) = +∞ if x = y
0
otherwise.
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f (x, y) = −1 if x = y
0
otherwise.
OOlivier Bernardi – p.9/37
Example
Hard particules in Ω = {1, . . . , q}.
φ(x, y) = +∞ if x = y
0
otherwise.
f (x, y) = −1 if x = y
0
otherwise.
Mayer’s weight of G :
X Y
e(G)q c(G)
W (G) =
f (xi , xj ) = (−1)
.
Ωn (i,j)∈G
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OOlivier Bernardi – p.9/37
Example
Hard particules in Ω = {1, . . . , q}.
φ(x, y) = +∞ if x = y
0
otherwise.
f (x, y) = −1 if x = y
0
otherwise.
Mayer’s weight of G :
X Y
e(G)q c(G)
W (G) =
f (xi , xj ) = (−1)
.
Ωn (i,j)∈G
Mayer’s correspondence
X
W (G) = n!Z(Ω, n) shows :
G⊆Kn
X
G⊆Kn
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(−1)e(G) q c(G)
q
= n!
= q(q − 1) . . . (q − n + 1).
n
Olivier Bernardi – p.9/37
Allowing any number of particules
The grand canonical partition function is
X
Zgr (Ω, T, z) =
Z(Ω, T, n)λn z n .
n
In terms of Mayer’s weights :
Zgr (Ω, T, z) =
X
n
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!
X W (G)z |G|
1 X
n n
W
(G)
λ
z =
.
n
λ n! G⊆K
|G|!
G
n
Olivier Bernardi – p.10/37
Pressure
The pressure of the system is given by
kT
P (Ω, T, z) =
log (Zgr (Ω, T, z)) .
|Ω|
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OOlivier Bernardi – p.11/37
Pressure
The pressure of the system is given by
kT
P (Ω, T, z) =
log (Zgr (Ω, T, z)) .
|Ω|
Since Mayers weights are multiplicative
kT
kT
P (Ω, T, z) =
log (Zgr (Ω, T, z)) =
|Ω|
|Ω|
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G
X
connected
W (G)z |G|
.
|G|!
Olivier Bernardi – p.11/37
Example
Hard particules in Ω = {1, . . . , q}.
Grand canonical partition function :
X
X q Zgr (Ω, T, z) =
Z(Ω, T, n)z n =
z n = (1 + z)q .
n
n
n
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OOlivier Bernardi – p.12/37
Example
Hard particules in Ω = {1, . . . , q}.
Grand canonical partition function :
X
X q Zgr (Ω, T, z) =
Z(Ω, T, n)z n =
z n = (1 + z)q .
n
n
n
Pressure :
kT
P (Ω, T, z) =
log (Zgr (Ω, T, z)) = kT log(1 + z).
|Ω|
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Olivier Bernardi – p.12/37
Example
Mayer’s weights : W (G) = (−1)e(G) q c(G) .
Pressure :
kT
P (Ω, T, z) =
|Ω|
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G
X
connected
W (G)z |G|
= kT
|G|!
G
X
connected
(−1)e(G) z |G|
.
|G|!
OOlivier Bernardi – p.13/37
Example
Mayer’s weights : W (G) = (−1)e(G) q c(G) .
Pressure :
kT
P (Ω, T, z) =
|Ω|
G
X
connected
W (G)z |G|
= kT
|G|!
G
X
connected
(−1)e(G) z |G|
.
|G|!
Comparing the two expressions of the pressure yields :
G
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X
connected
(−1)e(G) z |G|
= log(1 + z).
|G|!
OOlivier Bernardi – p.13/37
Example
Mayer’s weights : W (G) = (−1)e(G) q c(G) .
Pressure :
kT
P (Ω, T, z) =
|Ω|
G
X
connected
W (G)z |G|
= kT
|G|!
G
X
connected
(−1)e(G) z |G|
.
|G|!
Comparing the two expressions of the pressure yields :
G
X
connected
In other words :
G⊆Kn
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(−1)e(G) z |G|
= log(1 + z).
|G|!
X
(−1)e(G) = (−1)n−1 (n − 1)!.
connected
Olivier Bernardi – p.13/37
How did we get there ?
Mayer
Z(Ω, T, z)
X W (G)z |G|
|G|!
G
log
log
A
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=
B
Olivier Bernardi – p.14/37
A killing involution
G⊆Kn
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X
(−1)e(G) = (−1)n−1 (n − 1)!
connected
OOlivier Bernardi – p.15/37
A killing involution
G⊆Kn
X
(−1)e(G) = (−1)n−1 (n − 1)!
connected
We define an involution Φ on the set of connected graphs :
- Order the edges of Kn lexicographicaly.
- Define E ∗ (G) = {e = (i, j) / i and j are connected by G>e },
Φ(G) = G
G ⊕ min(E ∗ (G))
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if E ∗ (G)
=∅
otherwise.
OOlivier Bernardi – p.15/37
A killing involution
G⊆Kn
X
(−1)e(G) = (−1)n−1 (n − 1)!
connected
We define an involution Φ on the set of connected graphs :
- Order the edges of Kn lexicographicaly.
- Define E ∗ (G) = {e = (i, j) / i and j are connected by G>e },
Φ(G) = G
G ⊕ min(E ∗ (G))
if E ∗ (G)
=∅
otherwise.
Prop [B.] : The only remaining graphs are the increasing
spanning trees.
(Known to be in bijection with the permutations of
{1, .., n − 1}.)
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Olivier Bernardi – p.15/37
Increasing trees
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4
3
4
3
4
3
1
2
1
2
1
2
4
3
4
3
4
3
1
2
1
2
1
2
Olivier Bernardi – p.16/37
For those familiar with the Tutte Polynomial
The sum of the Mayer’s weight correspond to the
evaluations of TKn (1, 0).
This is the number of internal spanning trees.
Subgraph expansion ⇐⇒ Spanning tree expansion
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Olivier Bernardi – p.17/37
Hard-core continuum gas
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Olivier Bernardi – p.18/37
Hard-core continuum gas
Hard particules in Ω = [0, q].
Ω
x2
φ(x, y) = +∞ if |x − y| < 1
0
otherwise.
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x1
x3
f (x, y) = −1 if |x − y| < 1
0
otherwise.
OOlivier Bernardi – p.19/37
Hard-core continuum gas
Hard particules in Ω = [0, q].
x2
Ω
x1
x3
φ(x, y) = +∞ if |x − y| < 1
0
otherwise.
W (G) =
ZZ
Y
f (xi , xj )dx1 ..dxn .
Ωn (i,j)∈G
kT
P (Ω, T, z) =
|Ω|
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f (x, y) = −1 if |x − y| < 1
0
otherwise.
G
X
connected
W (G)z |G|
.
|G|!
Olivier Bernardi – p.19/37
Thermodynamical limit (|Ω| → ∞)
x2
x1
x3
P (T, z) ≡ lim P (Ω, T, z) = kT
|Ω|→∞
G
X
connected
W∞ (G)z |G|
|G|!
where,
WΩ (G)
=
W∞ (G) ≡ lim
|Ω|→∞
|Ω|
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ZZ
n−1 ;
Y
f (xi , xj )dx2 ..dxn .
x1 =0 (i,j)∈G
Olivier Bernardi – p.20/37
Mayer’s weight for the hard-core gas
f (x, y) = −1 if |x − y| < 1
0
otherwise.
W (G) =
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ZZ
n−1 ;
Y
f (xi , xj )dx2 ..dxn .
x1 =0 (i,j)∈G
OOlivier Bernardi – p.21/37
Mayer’s weight for the hard-core gas
f (x, y) = −1 if |x − y| < 1
0
otherwise.
W (G) =
ZZ
n−1 ;
Y
x1 =0 (i,j)∈G
P (T, z) = kT
G
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f (xi , xj )dx2 ..dxn .
X
connected
W (G)z |G|
|G|!
Olivier Bernardi – p.21/37
Mayer’s diagram for the hard-core gas
Mayer
Z(T, z)
X W (G)z |G|
G
log
(−1)
log
n−1 n−1
n
Cayley trees ! ?
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|G|!
=
P
G⊆Kn
W∞ (G)
[Labelle, Leroux, Ducharme : SLC 54]
Olivier Bernardi – p.22/37
Slicing W (G) [Lass]
W (G) =
ZZ
n−1 ;
f (xi , xj )dx2 ..dxn ,
x1 =0 (i,j)∈G
e(G)
× Volume(ΠG ),
\
is the polytope
|xi − xj | ≤ 1.
= (−1)
where ΠG ⊂ Rn−1
Y
(i,j)∈G
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OOlivier Bernardi – p.23/37
Slicing W (G) [Lass]
W (G) =
ZZ
n−1 ;
f (xi , xj )dx2 ..dxn ,
x1 =0 (i,j)∈G
e(G)
× Volume(ΠG ),
\
is the polytope
|xi − xj | ≤ 1.
= (−1)
where ΠG ⊂ Rn−1
Y
(i,j)∈G
Example :
G:
x1
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x3
ΠG :
x3
x2
x2
OOlivier Bernardi – p.23/37
Slicing W (G) [Lass]
W (G) =
ZZ
n−1 ;
f (xi , xj )dx2 ..dxn ,
x1 =0 (i,j)∈G
e(G)
× Volume(ΠG ),
\
is the polytope
|xi − xj | ≤ 1.
= (−1)
where ΠG ⊂ Rn−1
Y
(i,j)∈G
Example :
G:
x1
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x3
ΠG :
x3
x2
x2
Olivier Bernardi – p.23/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
3
2
h(xi ) = 1
xi
0
−1
=0
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(xi ) = 1
OOlivier Bernardi – p.24/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
|xi − xj | < 1 ⇐⇒
3
xj
2
1
xi
xj
xi
0
−1
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OOlivier Bernardi – p.24/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
Prop [Lass] : (x2 , .., xn ) ∈ ΠG ? only depends on the
integer parts h(x2 ), .., h(xn ) and the order of the fractional
parts (x2 ), .., (xn ).
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OOlivier Bernardi – p.24/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
Prop [Lass] : (x2 , .., xn ) ∈ ΠG ? only depends on the
integer parts h(x2 ), .., h(xn ) and the order of the fractional
parts (x2 ), .., (xn ).
h1
h2
h3
h4
h5
h6
=0
=1
=0
=2
= −1
=0
0 = 1 < 4 < 6 < 2 < 5 < 3
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3
2
1
x4
x2
x3
0 x x6
1
x5
−1
OOlivier Bernardi – p.24/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
Prop [Lass] : (x2 , .., xn ) ∈ ΠG ? only depends on the
integer parts h(x2 ), .., h(xn ) and the order of the fractional
parts (x2 ), .., (xn ).
3
x6
x1
x5
2
x4
x2
x3
x2
1
0
−1
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x4
x6
x1
x3
x5
OOlivier Bernardi – p.24/37
Slicing W (G) [Lass]
Fractional representation xi = h(xi ) + (xi ).
Prop [Lass] : (x2 , .., xn ) ∈ ΠG ? only depends on the
integer parts h(x2 ), .., h(xn ) and the order of the fractional
parts (x2 ), .., (xn ).
Each subpolytope ∆ defined by h2 , .., hn and an order on
1
(x2 ), .., (xn ) has volume
.
(n − 1)!
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Olivier Bernardi – p.24/37
Counting labelled schemes
x3
ΠG :
x3
G:
x1
x2
x2
x1
x1
x2 x3
x3 x2
x1
x2
x1
x3
x1
x2 x3
x1
x2
x3 x2
(−1)e(G)
Each labelled scheme has weight
.
(n − 1)!
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Olivier Bernardi – p.25/37
Rearanging the sum
X
W (G) =
X
(−1)e(G)
#{S labelled scheme containing G}
(n − 1)!
G⊆Kn
G⊆Kn
connected
connected
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OOlivier Bernardi – p.26/37
Rearanging the sum
X
W (G) =
X
(−1)e(G)
#{S labelled scheme containing G}
(n − 1)!
G⊆Kn
G⊆Kn
connected
connected
=
S
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X
labelled scheme
1
(n − 1)!
G
X
(−1)e(G)
contained in S
OOlivier Bernardi – p.26/37
Rearanging the sum
X
X
W (G) =
(−1)e(G)
#{S labelled scheme containing G}
(n − 1)!
G⊆Kn
G⊆Kn
connected
connected
=
S
labelled scheme
=
S
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X
X
scheme
G
1
(n − 1)!
X
G
X
(−1)e(G)
contained in S
(−1)e(G)
contained in S
Olivier Bernardi – p.26/37
Rearranging the sum
3
2
1
0
−1
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G
X
(−1)e(G)
contained in S
Olivier Bernardi – p.27/37
A killing involution
We define an involution Φ on the set of connected graphs
contained in S :
- Order the edges of Kn lexicographicaly.
- Define E ∗ (G) = {e = (i, j) / i and j are connected by G>e },
Φ(G) = G
G ⊕ min(E ∗ (G))
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if E ∗ (G)
=∅
otherwise.
OOlivier Bernardi – p.28/37
A killing involution
We define an involution Φ on the set of connected graphs
contained in S :
- Order the edges of Kn lexicographicaly.
- Define E ∗ (G) = {e = (i, j) / i and j are connected by G>e },
Φ(G) = G
G ⊕ min(E ∗ (G))
if E ∗ (G)
=∅
otherwise.
Proposition [B.] : The only remaining graphs are the
increasing spanning trees.
Corrolary :
X
G
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(−1)e(G) = (−1)n−1 #{increasing tree on S}.
contained in S
Olivier Bernardi – p.28/37
A killing involution
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OOlivier Bernardi – p.29/37
A killing involution
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Olivier Bernardi – p.29/37
Bijection with Cayley trees
Theorem [B.] :
S
[
{increasing tree} are in bijection with
scheme
rooted Cayley trees.
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OOlivier Bernardi – p.30/37
Bijection with Cayley trees
Theorem [B.] :
S
[
{increasing tree} are in bijection with
scheme
rooted Cayley trees.
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OOlivier Bernardi – p.30/37
Bijection with Cayley trees
Theorem [B.] :
S
[
{increasing tree} are in bijection with
scheme
rooted Cayley trees.
4
6
2
1
9
7
3
8
5
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OOlivier Bernardi – p.30/37
Bijection with Cayley trees
Theorem [B.] :
S
[
{increasing tree} are in bijection with
scheme
rooted Cayley trees.
4
6
2
1
9
7
3
8
5
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Olivier Bernardi – p.30/37
Bijection with Cayley trees
Corollary [B.] :
X
W (G) =
G⊆Kn
connected
S
X
X
(−1)e(G)
scheme G contained in S
= (−1)
n−1
S
X
#{increasing tree on S}
scheme
= (−1)n−1 nn−1 .
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Olivier Bernardi – p.31/37
Bijection with Cayley trees
1
2
1
3
3
3
2
1
2
1
3
1
3
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2
3
2
3
2
1
2
1
3
1
2
3
2
1
Olivier Bernardi – p.32/37
Concluding remarks
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Olivier Bernardi – p.33/37
Mayer’s transformation
Producing graph weights
Mayer
Z(Ω, T, z)
X W (G)z |G|
G
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|G|!
OOlivier Bernardi – p.34/37
Mayer’s transformation
Producing graph weights
Producing nasty identities
Mayer
Z(Ω, T, z)
X W (G)z |G|
|G|!
G
log
log
A
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=
B
Olivier Bernardi – p.34/37
Discrete hard-core gas (colorings)
Mayer
Z(Ω, T, z)
X W (G)z |G|
G
log
(−1)n−1 (n − 1)!
|G|!
log
=
X
(−1)e(G)
G⊆Kn
connected
CRM, Barcelona
OOlivier Bernardi – p.35/37
Discrete hard-core gas (colorings)
Potts model
⇐⇒ Subgraph expansion
Mayer
Z(Ω, T, z)
X W (G)z |G|
G
log
(−1)n−1 (n − 1)!
|G|!
log
=
X
(−1)e(G)
G⊆Kn
Spanning tree expansion
CRM, Barcelona
⇐⇒
connected
Subgraph expansion
Olivier Bernardi – p.35/37
Continuous hard-core gas
Mayer
Z(T, z)
X W (G)z |G|
G
log
(−1)
CRM, Barcelona
|G|!
log
n−1 n−1
n
=
P
G⊆Kn
W∞ (G)
Olivier Bernardi – p.36/37
Thanks.
CRM, Barcelona
Olivier Bernardi – p.37/37