Course X.V. IMSc 2017
Ch7 Heaps in statistical mechanics
For each item, the first number refers to the page of the slides, clicking on «video» takes you directly to the corresponding slide in the video. (to be completed)
Ch7a
2 March 2017
slides first part (pdf 23 Mo ) video
slides second part (pdf 18 Mo)
first part of slides
a few words about statistical mechanics 3
example 1: the Ising model 6
example 2: gas model 13
statistical mechanics 22
example 3: percolation 26
polyomino 29
animal 31
the directed animal problem 34
directed animal: definition 36
critical exponents for the length and the width 40
directed animals on a cylinder: the formula of Derrida, Nadal, Vannimenus 44
directed animals and heaps of dimers 47
bijection directed animal (square lattice) and strict pyramids of dimers 49
algebraic equations for directed animals 50, 51
Motzkin paths 52
algebraic equations for prefix of Motzkin paths 55, 56
random directed animals 61, 62
complements: compact source size directed animals 63
algebraic system of equations for compact source size directed animals 69
the formula 3^n 70
end of the first part of slides of Ch7a 73
second part of slides
directed animals on a triangular lattice 2
bijection directed animal (triangular lattice) and pyramids of dimers 4
directed animals on a bounded strip 6
generating function for directed animal on a bounded circular strip
and proof a Derrida, Nadal, Vannimenus formula 10
combinatorial understanding of the thermodynamic limit with 1D gas model 11
density of the gas: definition 17
combinatorial interpretation of the density 18
research problem: combinatorial interpretation of the partition function Z(t) 21
the hard hexagons gas model 23
interpretation of the density of the gas with pyramids of hexagons 29
combinatorial understanding of the thermodynamic limit 34
proof of the interpretation with pyramids of hexagons 37
a proposition related to the limit of the domain D 40
Baxter’s solution of the hard hexagons model 41
research problem about the hard hexagons partition function Z(t) 52
end of the second part of slides of Ch7a 57
Ch7b
13 March 2017
slides first part (pdf 22 Mo) video
slides second part (pdf 21 Mo)
first part of slides
from the previous lecture 3
algebricity of the density for hard hexagons 15
research problem (5+++) 16, 17
Lorentzian triangulations in 2D quantum gravity 20
a brief introduction to quantum gravity 21
classical ... 22
general relativity 24
the quantum world 27
string theory 32
Alain Connes non-commutative geometry 34
loop quantum gravity 35
causal sets 38
causal dynamical triangulations 39
end of the first part of slides of Ch7b 47
second part of slides
2D Lorentzian triangulations 3
path integral amplitude for the propagation of the geometry 11
(the four parameters for the enumeration of 2D Lorentzian triangulations)
Lorentzian triangulations on a cylinder 12
border conditions for Lorentzian triangulations 14, 15
bijection heaps of dimers Lorentzian triangulations 16
the four parameters generating function for Lorentzian triangulations with border conditions 26
bijection double semi-pyramids Lorentzian triangulations with left-right border conditions 36
exercise: bijection double semi-pyramids -- (general) heaps of dimers 37
the curvature parameter of the 2D space-time 41
interpretation of the up and down curvature on the heaps of dimers 46
an example with the stairs decomposition 47-53
characterization of heaps of dimers with zero up-curvature and zero total curvature 56-60
the nordic decomposition of a heap of dimers 61
connected heap of dimers 64
multi-directed animal (Bousquet-Mélou, Rechnitzer) 65-67
Bousquet-Mélou--Rechnitzer formula for connected heaps of dimers 70
bijective proof of this formula with the nordic decomposition of a connected heap 72-75
end of the bijective proof: Fibonacci polynomials and Catalan generating function 54
(solution of exercise Ch2b, p103)
application of the nordic decomposition for partially directed animals (Bacher) 91
extensions: Lorentzian quantum gravity in (1+1)+1 dimension 92
end of the second part of slides of Ch7b 99
Ch7c q-Bessel functions in physics
16 March 2017
slides first part (pdf 23 Mo) video
slides second part (pdf 15 Mo)
Epilogue slides (pdf 8 Mo)
first part of slides
Bessel functions and q-Bessel functions 2
from the previous lecture 6
parallelogram polyominoes (staircase polygons) and q-Bessel functions 12
the 3 parameters generating function 16
bijection parallelogram polyominoes -- semi-pyramids of segments 17
proof of the 3 parameters generating function for parallelogram polyominoes 31
from integers partitions to q-Bessel functions 35
q-Bessel functions as trivial heaps of segments 40
random parallelogram polyominoes 41
the Catalan garden 44
A festival of bijections 47
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with the stairs decomposition of a heap of dimers 49
bijection parallelogram polyominoes -- Dyck paths
bijection Dyck paths -- semi-pyramids of dimers 56
video with violin 57
bijection semi-pyramids of dimers -- semi-pyramids of segments 65
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with Lukasiewicz paths 69
bijection Dyck paths -- (reverse) Lukasiewicz paths 78
bijection (reverse) Lukasiewicz paths -- semi-pyramids of segments 83
other description of the bijection (parallelogram polyominoes -- semi-pyramids of segments)
with the bijection Psi (paths -- heaps of oriented loops + trail) 87
bijection Dyck paths -- heaps of oriented loops 89
end of the first part of slides of Ch7c 106
second part of slides
Complements: q-Bessel functions and SOS (Solid-on-Solid) model 3
definition of the SOS path 5
weight of the SOS path 6, 7
the 3 parameters generating function for SOS paths 8
from SOS paths to heaps of segments 11-14
an involution for the term x(1-y^2)
partially directed paths with interactions 16
particular case: weighted heaps of dimers and Ramanujan continued fraction 17
area: q-Catalan 19
Rogers-Ramanujan identities 23
D-partitions 26
from partitions to D-partitions 27-30
generating function for weighted semi-pyramids of dimers 33
interpretations of the Rogers-Ramanujan identities with intergers partitions 35, 36
Ramanujan continued fraction 37
interpretation of Ramanujan continued fraction as weighted semi-pyramids of dimers 38
decomposition of a semi-pyramids as sequence of primitive semi-pyramids 40-53
end of the proof 56
Ramanujan continued fraction as the ratio N/D 58
back to the system of q-equations for the partition function Z(t) of the hard gas model 59
Andrews ‘s interpretation of the reciprocal of Ramanujan identities 60
other future chapters 66
end of the second part of slides of Ch7c 70
Epilogue: Kepler towers
Kepler towers 2
definition: system of Kepler towers 4
proposition: enumeration of systems of Kepler towers with Catalan numbers 11
Kepler disks 13
why Kepler towers ? 15
Kepler mysterium cosmographicum 19
Kepler towers and Strahler number of a binary tree 23
logarithmic height of a Dyck path 26
Programs to read by D. Knuth 29
Many Thanks 30
the end of the Epilogue and the end of the course ! 31
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the IMSc 2016 bijective course website
