158
On exact computation of square ice entropy Silv` ere Gangloff LIP, ENS Lyon March 11, 2019 1 / 32
On exact computation of square ice entropy
Silvere Gangloff
LIP, ENS Lyon
March 11, 2019
1 / 32
I. Representations of square ice
2 / 32
Square ice model [Pauling-Lieb]:
3 / 32
Wang tiles representation [Six-vertex model]
4 / 32
Wang tiles representation [Six-vertex model]
4 / 32
Wang tiles representation [Six-vertex model]
4 / 32
Wang tiles representation [Six-vertex model]
5 / 32
Wang tiles representation [Six-vertex model]
5 / 32
Discrete curves subshift [X s]:
6 / 32
Discrete curves subshift [X s]:
6 / 32
Example of apparition in symbolic dynamics:
S.Gangloff, M. Sablik, Quantified block gluing, aperiodicity and entropy ofmultidimensional SFT , 2017.
Entropy value ?
E.H. Lieb, Residual entropy of square ice, Physical Review, 1967.
→ Proof under some hypothesis
S. Gangloff, A proof that square ice entropy is 32 log2(4/3), 2019.
7 / 32
Example of apparition in symbolic dynamics:
S.Gangloff, M. Sablik, Quantified block gluing, aperiodicity and entropy ofmultidimensional SFT , 2017.
Entropy value ?
E.H. Lieb, Residual entropy of square ice, Physical Review, 1967.
→ Proof under some hypothesis
S. Gangloff, A proof that square ice entropy is 32 log2(4/3), 2019.
7 / 32
Example of apparition in symbolic dynamics:
S.Gangloff, M. Sablik, Quantified block gluing, aperiodicity and entropy ofmultidimensional SFT , 2017.
Entropy value ?
E.H. Lieb, Residual entropy of square ice, Physical Review, 1967.
→ Proof under some hypothesis
S. Gangloff, A proof that square ice entropy is 32 log2(4/3), 2019.
7 / 32
Example of apparition in symbolic dynamics:
S.Gangloff, M. Sablik, Quantified block gluing, aperiodicity and entropy ofmultidimensional SFT , 2017.
Entropy value ?
E.H. Lieb, Residual entropy of square ice, Physical Review, 1967.
→ Proof under some hypothesis
S. Gangloff, A proof that square ice entropy is 32 log2(4/3), 2019.
7 / 32
II. Subshifts of finite type and entropy
8 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
1 01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1
01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops
0
1
1
9 / 32
SFT (subshift of finite type): subset of AZ2, defined by a finite set of
forbidden patterns.
Ex: Hard square shift, or hard core model.
Forbidden patterns 11
et 1 1 .
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
10
1 01
1
0
1 oops0
1
1
9 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy: ”quantity of possible states of the system”.
NN(X ): number of size N square patterns observable in the system.
Free tiles
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
1 1
0 0
0 1
0 1
0 0
1 1
1 0
1 0
1 1
1 0
1 1
0 1
1 0
1 1
0 1
1 1
1 1
1 1
N2(X ) = 222
NN(X ) = 2N2
Hard core
0 0
0 0
1 0
0 0
0 1
0 0
0 0
1 0
0 0
0 1
1 0
0 1
0 1
1 0
N2(X ) = 7
NN(X ) = 2N2(h(X )+o(1))
10 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2
= infN
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1
N = 2h(X )
r2N = 3
h(X )
r3N = 4
h(X )
r4
11 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2= inf
N
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1
N = 2h(X )
r2N = 3
h(X )
r3N = 4
h(X )
r4
11 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2= inf
N
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1
N = 2h(X )
r2N = 3
h(X )
r3N = 4
h(X )
r4
11 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2= inf
N
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1N = 2
h(X )
r2
N = 3h(X )
r3N = 4
h(X )
r4
11 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2= inf
N
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1N = 2
h(X )
r2N = 3
h(X )
r3
N = 4h(X )
r4
11 / 32
Entropy of a SFT X:
h(X ) = infN
log2(NN(X ))
N2= inf
N
log2(N locN (X ))
N2
Free tiles
h = 1
Hard core
h ≥ 1/2
Square ice [Lieb 67]
h = 32 log2(4/3)
Computable from above:
AlgorithmN = 1h(X )
r1N = 2
h(X )
r2N = 3
h(X )
r3N = 4
h(X )
r4
11 / 32
II. Lieb transfer matrices approach
12 / 32
Entropy of square ice:
h(X s) = limM,N
log2(NM,N(X s))
MN.
Stripes subshifts:
X s
X sN
h(X s) = limN
h(X sN)
N
13 / 32
Cylindric stripes subshifts:
X sN
XsN
Symmetry properties of square ice imply:
h(X s) = limN
h(XsN)
N
14 / 32
Cylindric stripes subshifts:
X sN
XsN
Symmetry properties of square ice imply:
h(X s) = limN
h(XsN)
N
14 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Symmetry properties of square ice:
15 / 32
Lieb transfer matrices:
VN(t)
≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t)
≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Lieb transfer matrices:
VN(t) ≡
R[w ]u
v
w
VN(t)[u, v ] =∑
uR[w ]v
t |w |.
where |w | = # of and
h(X s) = limN
log2(λmax(VN(1)))
N
16 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
LiebT xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
Lieb
T xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
Lieb
T xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
LiebT xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
LiebT xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
LiebT xN(t)
17 / 32
Computing maximal eigenvalue of VN(1), strategy:
VN(1)
VN(t)
Yang-Baxter
Analycity
Alg. Bethe ansatz
LiebT xN(t)
17 / 32
III. Yang-Baxter transfer matrices
18 / 32
R-matrices and monodromy matrices:
→ : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )
R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )
R(0, 1) =
(0
)
R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)
R(0, 1) =
(0 λ
)
R(0, 1) =
(0 λ0
)R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)
R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)
1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)
1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0
. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0
. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0
. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
R-matrices and monodromy matrices: → : input/output:
1
0
R(0, 1) =
( )R(0, 1) =
(0
)R(0, 1) =
(0 λ
)R(0, 1) =
(0 λ0
)
R(0, 1) =
(0 λ0 0
)1 0
0
0
0
0
1
0
0
1
1
0
. . .
. . .
η ∈ {0, 1}N
ε ∈ {0, 1}N
MN(1, 0)[ε,η]
Yang-Baxter transfer matrices:
TN [ε,η] =∑
u∈{0,1}
MN(u, u)[ε,η].
19 / 32
Composition of these matrices and condition for commutation:
0
1
0
1
1
0
Yang-Baxter equation:
R
R’S =
R’
RS
20 / 32
Composition of these matrices and condition for commutation:
0
1
0
1
1
0
Yang-Baxter equation:
R
R’S =
R’
RS
20 / 32
Trigonometric R-matrices: for t ∈ (0,√
2], 2− t2 = − cos(µt):
Rxµt =
1
sin(µt/2)
sin(µt − x) 0 0 0
0 sin(x) sin(µt) 00 sin(µt) sin(x) 00 0 0 sin(µt − x)
.
Bethe ansatz: if (pj)j is solution of:
Npj = 2πj − (n + 1)π −n∑
k=1
Θt(pj , pk)
then we have a candidate eigenvector for the eigenvalue:
n∏k=1
Lt(eipk ) +
n∏k=1
Mt(eipk ).
→ Known: existence and analycity in t.
21 / 32
Trigonometric R-matrices: for t ∈ (0,√
2], 2− t2 = − cos(µt):
Rxµt =
1
sin(µt/2)
sin(µt − x) 0 0 0
0 sin(x) sin(µt) 00 sin(µt) sin(x) 00 0 0 sin(µt − x)
.
Bethe ansatz: if (pj)j is solution of:
Npj = 2πj − (n + 1)π −n∑
k=1
Θt(pj , pk)
then we have a candidate eigenvector for the eigenvalue:
n∏k=1
Lt(eipk ) +
n∏k=1
Mt(eipk ).
→ Known: existence and analycity in t.
21 / 32
Trigonometric R-matrices: for t ∈ (0,√
2], 2− t2 = − cos(µt):
Rxµt =
1
sin(µt/2)
sin(µt − x) 0 0 0
0 sin(x) sin(µt) 00 sin(µt) sin(x) 00 0 0 sin(µt − x)
.
Bethe ansatz: if (pj)j is solution of:
Npj = 2πj − (n + 1)π −n∑
k=1
Θt(pj , pk)
then we have a candidate eigenvector for the eigenvalue:
n∏k=1
Lt(eipk ) +
n∏k=1
Mt(eipk ).
→ Known: existence and analycity in t.
21 / 32
Trigonometric R-matrices: for t ∈ (0,√
2], 2− t2 = − cos(µt):
Rxµt =
1
sin(µt/2)
sin(µt − x) 0 0 0
0 sin(x) sin(µt) 00 sin(µt) sin(x) 00 0 0 sin(µt − x)
.
Bethe ansatz: if (pj)j is solution of:
Npj = 2πj − (n + 1)π −n∑
k=1
Θt(pj , pk)
then we have a candidate eigenvector for the eigenvalue:
n∏k=1
Lt(eipk ) +
n∏k=1
Mt(eipk ).
→ Known: existence and analycity in t.
21 / 32
Trigonometric R-matrices: for t ∈ (0,√
2], 2− t2 = − cos(µt):
Rxµt =
1
sin(µt/2)
sin(µt − x) 0 0 0
0 sin(x) sin(µt) 00 sin(µt) sin(x) 00 0 0 sin(µt − x)
.
Bethe ansatz: if (pj)j is solution of:
Npj = 2πj − (n + 1)π −n∑
k=1
Θt(pj , pk)
then we have a candidate eigenvector for the eigenvalue:
n∏k=1
Lt(eipk ) +
n∏k=1
Mt(eipk ).
→ Known: existence and analycity in t.21 / 32
Identification to λmax(VN(t)): ingredients:
1 Simplification of equations when t =√
2.
2 For some HN diagonalised, VN(√
2)HN = HNVN(√
2).
3 Perron-Frobenius theorem.
4 Indentification around√
2, on (0,√
2) by analycity.
22 / 32
Identification to λmax(VN(t)): ingredients:
1 Simplification of equations when t =√
2.
2 For some HN diagonalised, VN(√
2)HN = HNVN(√
2).
3 Perron-Frobenius theorem.
4 Indentification around√
2, on (0,√
2) by analycity.
22 / 32
Identification to λmax(VN(t)): ingredients:
1 Simplification of equations when t =√
2.
2 For some HN diagonalised, VN(√
2)HN = HNVN(√
2).
3 Perron-Frobenius theorem.
4 Indentification around√
2, on (0,√
2) by analycity.
22 / 32
Identification to λmax(VN(t)): ingredients:
1 Simplification of equations when t =√
2.
2 For some HN diagonalised, VN(√
2)HN = HNVN(√
2).
3 Perron-Frobenius theorem.
4 Indentification around√
2, on (0,√
2) by analycity.
22 / 32
Identification to λmax(VN(t)): ingredients:
1 Simplification of equations when t =√
2.
2 For some HN diagonalised, VN(√
2)HN = HNVN(√
2).
3 Perron-Frobenius theorem.
4 Indentification around√
2, on (0,√
2) by analycity.
22 / 32
IV. Asymptotics
23 / 32
Asymptotics of counting functions:
nk/Nk → 1/2. Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Asymptotics of counting functions: nk/Nk → 1/2.
Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Asymptotics of counting functions: nk/Nk → 1/2. Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Asymptotics of counting functions: nk/Nk → 1/2. Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Asymptotics of counting functions: nk/Nk → 1/2. Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Asymptotics of counting functions: nk/Nk → 1/2. Solution of the
equations for Nk , nk : (p(k))j )j = (κt(α
(k)j ))j .
ξ(k)t : α 7→ 1
2πκt(α) +
nk + 1
2Nk+
1
2πNk
nk∑j=1
θt(α,α(k)j )
j/Nk
α(k)j
limN
log2(λmax(VN(1))
N= lim
k
1
Nk
nk∑j=1
f (α(k)j ) =
∫Rf (α)ρt(α)dα.
24 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
RR
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)t
ξ(0)t |Kξ(1)t |Kξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)t
ξ(0)t |K
ξ(1)t |Kξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |K
ξ(1)t |K
ξ(2)t |Kξ
(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |K
ξ(2)t |K
ξ(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |K
ξ(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |K
ξ(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |K
ξ(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Strategy:
1 Extend ξ(k)t on a stripe including R:
R
R
ξ(k)tξ
(0)t |Kξ(1)t |Kξ(2)t |K
ξ(+∞)t |K
2 Assume (ξt)ν(k) → ξ
(+∞)t on any compact K .
3 ξ(+∞)t verifies an integral equation with unique solution ρt .
4 Thus, ξ(k)t → ξ
(∞)t .
25 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Rarefaction of the roots and ξ(k)t biholomorphisms: ε > 0:
(ξ
(∞)t
)′> 0
1Nk
∣∣∣∣j : {α(k)j /∈
(ξ
(k)t
)−1([−M,M])}
∣∣∣∣ ≤ ε
-M M
Γ
V
The functions have distinct values on V and Γ. Thus they arebihilomorphisms onto V.
26 / 32
Lace integral expression of ξ(k)t :
Γ
j/Nk
By residues theorem:
ξ(k)t (α) =
1
2πκt(α) +
nk + 1
2Nk+
∮Γ
θt
((ξ
(k)t
)−1(α)
)e2iπsNk
e2iπsNk − 1ds + O(ε).
27 / 32
Lace integral expression of ξ(k)t :
Γ
j/Nk
By residues theorem:
ξ(k)t (α) =
1
2πκt(α) +
nk + 1
2Nk+
∮Γ
θt
((ξ
(k)t
)−1(α)
)e2iπsNk
e2iπsNk − 1ds + O(ε).
27 / 32
Fredholm integral equation: Limit and change of variable:
ξ(∞)t (α) =
1
2πκt(α) +
1
4+
∫ +∞
0θt(α)
(ξ
(∞)t
)′(α)dα.
Solution by Fourier transforms.
28 / 32
Fredholm integral equation: Limit and change of variable:
ξ(∞)t (α) =
1
2πκt(α) +
1
4+
∫ +∞
0θt(α)
(ξ
(∞)t
)′(α)dα.
Solution by Fourier transforms.
28 / 32
Final computation:
h(X s) =
∫R
log2(2| sin(κt(α))/2|).ρt(α)dα.
Through an expression of ρt =(ξ
(∞)t
)′and lace integrals computations:
h(X s) =3
2log2(4/3) .
29 / 32
Final computation:
h(X s) =
∫R
log2(2| sin(κt(α))/2|).ρt(α)dα.
Through an expression of ρt =(ξ
(∞)t
)′and lace integrals computations:
h(X s) =3
2log2(4/3) .
29 / 32
Final computation:
h(X s) =
∫R
log2(2| sin(κt(α))/2|).ρt(α)dα.
Through an expression of ρt =(ξ
(∞)t
)′and lace integrals computations:
h(X s) =3
2log2(4/3) .
29 / 32
V. Comments
30 / 32
Why mathematical physics are hard to read for mathematicians?
Archaeology of Knowledge,1969
Concept of discursive formation
Mathematics and mathematical physics are distinct discursive formations ;different conceptions of units of meaning, etc.
31 / 32
Further research:
1 Extensions: eight-vertex model [Baxter], dimer model [Lieb]..
2 Hard core model ? Tridimensional ice ? Kari-Culik tilings ?
3 Transformation of entropy by subshifts operators ?
32 / 32
Further research:
1 Extensions: eight-vertex model [Baxter], dimer model [Lieb]..
2 Hard core model ? Tridimensional ice ? Kari-Culik tilings ?
3 Transformation of entropy by subshifts operators ?
32 / 32
Further research:
1 Extensions: eight-vertex model [Baxter], dimer model [Lieb]..
2 Hard core model ? Tridimensional ice ? Kari-Culik tilings ?
3 Transformation of entropy by subshifts operators ?
32 / 32
