XAVIER VIENNOT
  • Foreword
    • Preface
    • Introduction
    • Acknowledgements
    • Lectures for wide audience
  • PART I
    • Preface
    • Abstract
    • Contents
    • Ch0 Introduction to the course
    • Ch1 Ordinary generating functions
    • Ch2 The Catalan garden
    • Ch3 Exponential structures and genarating functions
    • Ch4 The n! garden
    • Ch5 Tilings, determinants and non-intersecting paths
    • Lectures related to the course
    • List of bijections
    • Index
  • PART II
    • Preface
    • Abstract
    • Contents
    • Ch1 Commutations and heaps of pieces: basic definitions
    • Ch2 Generating functions of heaps of pieces
    • Ch3 Heaps and paths, flows and rearrangements monoids
    • Ch4 Linear algebra revisited with heaps of pieces
    • Ch5 Heaps and algebraic graph theory
    • Ch6 Heaps and Coxeter groups
    • Ch7 Heaps in statistical mechanics
    • Lectures related to the course
  • PART III
    • Preface
    • Abstract
    • Contents
    • Ch0 overview of the course
    • Ch1 RSK The Robinson-Schensted-Knuth correspondence
    • Ch2 Quadratic algebra, Q-tableaux and planar automata
    • Ch3 Tableaux for the PASEP quadratic algebra
    • Ch4 Trees and tableaux
    • Ch5 Tableaux and orthogonal polynomials
    • Ch6 Extensions: tableaux for the 2-PASEP quadratic algebra
    • Lectures related to the course
    • References, comments and historical notes
  • PART IV
    • Preface
    • Introduction
    • Contents
    • Ch0 Overview of the course
    • Ch1 Paths and moments
    • Ch2 Moments and histories
    • Ch3 Continued fractions
    • Ch4 Computation of the coefficients b(k) lambda(k)
    • Ch5 Orthogonality and exponential structures
    • Ch6 q-analogues
    • Lectures related to the course
    • Complements
    • References
  • Epilogue

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