The Art of Bijective Combinatorics Part II
Commutations and heaps of pieces
with interactions in physics, mathematics and computer science
The Institute of Mathematical Sciences, Chennai, India (January-March 2017)
Ch 7 Heaps in statistical mechanics
Ch 7a Ising model, gas model, directed animals and heaps of dimers, the formula 3^n, themodynamic limit, hard hexagons gas model
2 March 2017
slides_Ch7a (pdf 38 Mo )
video Ch7a link to YouTube
video Ch7a link to bilibili
a few words about statistical mechanics 3 0:19
example 1: the Ising model 6 4:36
example 2: gas model 13 10:34
discussion: why 3 colours ? 15:51
statistical mechanics 22 17:12
example 3: percolation 26 20:14
polyomino 29 21:25
animal 31 21:44
the directed animal problem 34 22:53
directed animal: definition 36 23:15
critical exponents for the length and the width 40 27:59
directed animals on a cylinder: the formula of Derrida, Nadal, Vannimenus 44 31:09
directed animals and heaps of dimers 47 34:40
bijection directed animal (square lattice) and strict pyramids of dimers 48, 49 34:44
discussion (about heaps, pre-heaps ...) 35:21
algebraic equations for directed animals 50, 51 35:59
Motzkin paths 52 37:27
algebraic equations for prefix of Motzkin paths 55, 56 38:09
formula for the number of directed animals 59 40:35
image of random directed animals 61, 62 42:48
complements: compact source size directed animals 63 43:44
the formula 3^n 66 44:56
algebraic system of equations for compact source size directed animals 69 46:26
two parameters (lower-width and number of source points) 71, 72 47:17
random compact source size directed animals 73 48:40
directed animals on a triangular lattice 74 49:19
bijection directed animal (triangular lattice) and pyramids of dimers 75 49:22
directed animals on a bounded strip 78 52:56
generating function for directed animal on a bounded circular strip
and proof a Derrida, Nadal, Vannimenus formula 81, 82 53:20
combinatorial understanding of the thermodynamic limit with 1D gas model 83 56:09
density of the gas: definition 89 1:03:01
combinatorial interpretation of the density 90 1:04:18
research problem: combinatorial interpretation of the partition function Z(t) 93 1:09:55
the hard hexagons gas model 95 1:13:44
interpretation of the density of the gas with pyramids of hexagons 101 1:14:51
combinatorial understanding of the thermodynamic limit 106 1:16:52
proof of the interpretation with pyramids of hexagons 109 1:17:04
a proposition related to the limit of the domain D 112 1:19:34
Baxter’s solution of the hard hexagons model 113 1:20:51
research problem about the hard hexagons partition function Z(t) 124 1:25:28
Hard core lattice gas model, hard square ? 127 1:28:27
the end of Ch7a 129 1:29:11
Ch 7b Lorenztian triangulation in 2D quantum gravity, the curvature parameter of the 2D space-time, connected heaps of dimers,
the nordic decomposition
13 March 2017
slides_Ch7b (pdf 40 Mo)
video Ch7b link to YouTube
video Ch7b link to bilibili
from the previous lecture 3 0:07
algebricity of the density for hard hexagons 15 10:06
research problem (5+++) 16, 17
Lorentzian triangulations in 2D quantum gravity 20 17:21
a brief introduction to quantum gravity 21 17:35
classical ... 22 17:47
general relativity 24 18:07
the quantum world 27 19:18
string theory 32 21:05
Alain Connes non-commutative geometry 34 22:02
loop quantum gravity 35 22:36
causal sets 38 23:15
causal dynamical triangulations 39 25:02
2D Lorentzian triangulations 47 27:09
path integral amplitude for the propagation of the geometry 55 30:09
(the four parameters for the enumeration of 2D Lorentzian triangulations)
Lorentzian triangulations on a cylinder 56 30:58
border conditions for Lorentzian triangulations 58, 59 31:38
bijection heaps of dimers Lorentzian triangulations 60 32:17
the four parameters generating function for Lorentzian triangulations with border conditions 70 35:56
exercise: 3-parameters generating function for pyramids of dimers 74, 75 38:39
expression for the 4-parameters generating function for pyramids of dimers 77, 78 42:00
second proof of the 4-parameters formula 79 44:16
bijection double semi-pyramids Lorentzian -- triangulations with left-right border conditions 80 45:18
exercise: bijection double semi-pyramids -- (general) heaps of dimers 81 45:49
the curvature parameter of the 2D space-time 85 48:44
interpretation of the up and down curvature on the heaps of dimers 90 50:26
an example with the stairs decomposition 91-97 51:14
characterization of heaps of dimers with zero up-curvature and zero total curvature 101-104 57:24
the nordic decomposition of a heap of dimers 105 59:20
connected heap of dimers 108 1:01:21
multi-directed animal (Bousquet-Mélou, Rechnitzer) 109-111 1:03:34
Bousquet-Mélou--Rechnitzer formula for connected heaps of dimers 113, 114 1:05:51
bijective proof of this formula with the nordic decomposition of a connected heap 116-119 1:07:35
end of the bijective proof: Fibonacci polynomials and Catalan generating function 128 1:17:42
(solution of exercise Ch2b, p103)
application of the nordic decomposition for partially directed animals (Bacher) 135 1:20:39
extensions: Lorentzian quantum gravity in (1+1)+1 dimension 136 1:21:39
the end 143 1:22:22
Ch 7c q-Bessel functions in physics
parallelogram polyominoes and q-Bessel, other description of the bijection parallelogram polyominoes -- semi-pyramids of segments,
q-Bessel and SOS model, Ramanujan continued fraction and heaps of dimers
16 March 2017
slides _Ch7c (pdf 35 Mo)
Epilogue slides (pdf 8 Mo)
video Ch7c link to YouTube
video Ch7c link to bilibili
Bessel functions and q-Bessel functions 3 0:13
from the previous lecture 6 1:52
parallelogram polyominoes (staircase polygons) and q-Bessel functions 12 4:18
the 3 parameters generating function 16 6:52
bijection parallelogram polyominoes -- semi-pyramids of segments 17 8:43
proof of the 3 parameters generating function for parallelogram polyominoes 31 13:56
from integers partitions to q-Bessel functions 35 15:09
q-Bessel functions as trivial heaps of segments 40 18:10
random parallelogram polyominoes 41 19:07
the Catalan garden 44 20:59
A festival of bijections 47 21:40
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with the stairs decomposition of a heap of dimers 48 21:45
bijection staircase polygons (parallelogram polyominoes) -- Dyck paths 49 21:50
bijection Dyck paths -- semi-pyramids of dimers 56 23:19
video with violin 57 23:53
stair decomposition 63 28:05
bijection semi-pyramids of dimers -- semi-pyramids of segments 65 28:16
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with Lukasiewicz paths 69 29:16
bijection Dyck paths -- (reverse) Lukasiewicz paths 78 30:38
bijection (reverse) Lukasiewicz paths -- semi-pyramids of segments 83 31:21
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with the bijection Psi (paths -- heaps of oriented loops + trail) 87 32:03
bijection Dyck paths -- heaps of oriented loops 89 32:51
a festival of bijections 105 34:28
Complements: q-Bessel functions and SOS (Solid-on-Solid) model 106 34:52
definition of the SOS path 108 35:29
weight of the SOS path 109, 110 35:53
the 3 parameters generating function for SOS paths 111 37:08
from SOS paths to heaps of segments 114-117 38:33
an involution for the term x(1-y^2) 118 41:00
partially directed paths with interactions 119 41:54
particular case: weighted heaps of dimers and Ramanujan continued fraction 120 42:48
area: q-Catalan 122 43:31
Rogers-Ramanujan identities 126 45:39
D-partitions 129 46:07
from partitions to D-partitions 130-133 46:31
generating function for weighted semi-pyramids of dimers 136 49:44
interpretations of the Rogers-Ramanujan identities with intergers partitions 138, 139 50:36
Ramanujan continued fraction 140 52:07
interpretation of Ramanujan continued fraction as weighted semi-pyramids of dimers 141 52:09
decomposition of a semi-pyramids as sequence of primitive semi-pyramids 143-156 52:59
end of the proof 159 54:18
Ramanujan continued fraction as the ratio N/D 161 54:49
back to the system of q-equations for the partition function Z(t) of the hard gas model 162 55:53
Andrews ‘s interpretation of the reciprocal of Ramanujan identities 163 57:24
other future chapters 169 1:00:31
the end of Ch7c 173 1:10:04
Epilogue: Kepler towers
16 March 2017
slides Epilogue (pdf 8 Mo)
video (end of video of Ch 7c)
Kepler towers 2 1:10:09
definition: system of Kepler towers 4 1:10:58
proposition: enumeration of systems of Kepler towers with Catalan numbers 11 1:14:08
Kepler disks 13 1:15:19
why Kepler towers ? 15 1:15:41
Kepler mysterium cosmographicum 19 1:16:47
Kepler towers and Strahler number of a binary tree 23 1:17:29
logarithmic height of a Dyck path 26 1:18:28
Programs to read by D. Knuth 29 1:20:50
Many Thanks 30 1:21:20
the end of the Epilogue and the end of the course ! 31 1:21:59