Escaping the branched polymer phasein dynamical triangulations

Luca Lionni
YITP – Kyoto U

10/09/18 – Tohoku U – Risan workshop

A simple combinatorial problem in discrete geometry

A simple combinatorial problem in discrete geometry
In any dimension, take a collec5on of “rigid” building blocks of your choice.
Glue them together in every possible way.

1

A simple combinatorial problem in discrete geometry
In any dimension, take a collec5on of “rigid” building blocks of your choice.
Glue them together in every possible way.

à Can we count the resul5ng discrete
spaces according to their global
curvature?

1

A simple combinatorial problem in discrete geometry
In any dimension, take a collec5on of “rigid” building blocks of your choice.
Glue them together in every possible way.

à Can we count the resul5ng discrete
spaces according to their global
curvature?
à What are the asympto5c proper5esof the spaces of maximal curvature?
What do they look like in thecon5nuum?

11 – Curvature…?
2 –Discrete quantum gravity (Dynamical triangula5ons)
3 – Previously known results
4 – Some recent results
5 – Random tensor models
6 – Ongoing work

1 – Curvature…?

1 – Curvature…?

Start with the simplest case: triangula5ons

l2

1 – Curvature…?

In a triangula5on, suppose that all edges have the same lengthà “canonical geometry” (no5on of distance, local curvature…)l2

1 – Curvature…?
Local curvature:
Number of D-simplices around(D-2)-simplices3

1 – Curvature…?
Local curvature:

Dimension D=2

Number of D-simplices around(D-2)-simplices

Number of equilateral triangles aroundver5ces :•posi5ve local
curvature•

locally ﬂat•

nega5ve local
curvatureDeﬁcit angle in dimension 22⇡ 1

N (v)
63

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I!"#JV#$%&'$)(#"!
!
!
"#$%&'!()*+#!&)!,&-#).&/)!"!
"1"2⇡
ND (vD 2 ) ⇥ arccos"D""

G$%(,#$*,%!$&%"

U;%+(/"*8".(./-<(0/-"-/*;,0"(01(#"S""arccos 13M"

1 – Curvature…?
Total curvature:↵ = arccos

Curv ⇠X⇣2⇡

vD2⇠ 2⇡nD 2↵ ND (vD 2 )

1D⌘

D(D + 1)nD↵
2

à Computed from the number of (D-2)-simplices and D-simplices ( nD

2 and nD )5

1 – Curvature…?
Total curvature:↵ = arccos

Curv ⇠X⇣2⇡

vD2⇠ 2⇡nD 2↵ ND (vD 2 )

1D⌘

D(D + 1)nD↵
2

à Computed from the number of (D-2)-simplices and D-simplices ( nD
à Maximize the curvature at ﬁxed nD

2 and nD )Maximize nD 2 at ﬁxed nD
51 – Curvature…?
Dimension D=2: Bigger polygons???

6

1 – Curvature…?
Dimension D=2: Bigger polygons???

A possible choice (only depends onone length)à6

1 – Curvature…?
Dimension D=2: Bigger polygons???

A possible choice (only depends onone length)àOther possible choice which works in what follows: 1 choice of angle for each polygon

61 – Curvature…?
Dimension D: Bigger polygons???

0àA possible choice (only depends onone length): SAME07

2 - Discrete quantum gravity:
Dynamical triangula5ons from bigger building blocks…

2 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =ZM

D[g]eRD

d x

Manifoldp|g|(2⇤116⇡G R)

Ricci scalar curvature

MetricCosmological constantNewton’s constantà (Euclidean) general rela5vity without mafer from least ac5on principle8

2 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =ZMD[g]eR• Complicated object• Not well deﬁned

Dd xp|g|(2⇤116⇡G R)

à Try to make sense of it by consideringa discrete analog

92 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =Z

M

D[g]eRDd xp|g|(2⇤116⇡G R)

Edges all have same length l(canonical geometry)(David, Ambjorn, Kazakov and many more… 80’s)!

XT
connectedtriangulationof M

1e SdiscreteCTl10

1 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =Z

M

D[g]eRDd xp|g|(2⇤116⇡G R)!

XT
connected
triangulationof M

1e Sdiscrete
CTAllow topology ﬂuctua5ons at microscopic level11

1 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =Z

M

D[g]eR

Dd xp|g|(2⇤116⇡G R)!

XT
connected
triangulation

1e SdiscreteCTl

12

1 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =Z

D[g]eMRDd xp|g|(2⇤116⇡G R)!

T
connected
triangulation(Regge)Sdiscrete ! D ⇥ nD (T )X 1e  D nD e  D 2 nD 2CT

D 2 ⇥ nD 2 (T )l# of D-simplices# of (D-2)-simplices

12

1 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension D

F =Z

D[g]eM

RDd x

p|g|(2⇤116⇡G R)!

T
connected
triangulation(Regge)Sdiscrete ! D ⇥ nD (T )ZX 1e  D nD e  D 2 nD 2CT

D 2 ⇥ nD 2 (T )lp
d x |g|R / Curvature
DMCurvature

number of (D-2) and D-simplices

13

1 – Discrete quantum gravity
Einstein-Hilbert par55on func5on for Euclidean pure gravity in dimension DF =Z

D[g]eM

RDd xp|g|(2⇤116⇡G R)!

T
connected
triangulation(Regge)Sdiscrete ! D ⇥ nD (T )X 1e  D nD e  D 2 nD 2CT

D 2 ⇥ nD 2 (T )l1>> 1
D 2 /G13

Quantum gravity? Analogy with thermodynamics…
Microscopic descrip4on of gaz

Microscopic descrip4on of space-4me

Accessible state S

Triangula5on STn(S) par5clesnD (ST ) D-simplices

n ! +1Dist ! 0
Grand-canonical par55on func5onCon5nuum limit

Thermodynamical limit

XE(S)eµn(S) e

states S

Inverse temperature
Chemical poten5alEnergy

=1
kB T

nD ! +1
l !0

Discrete Einstein-Hilbert par55on func5onXe D nD (ST ) eD 2 nD 2 (ST )STconnectedtriangulation

Inverse of Newton constantµD ?

E(S)nD 2 ?

D 2 /1G

14

Quantum gravity? Analogy with thermodynamics…
Microscopic descrip4on of gaz

Microscopic descrip4on of space-4me

Accessible state S

Triangula5on STn(S) par5clesnD (ST ) D-simplices

n ! +1Dist ! 0
Grand-canonical par55on func5onCon5nuum limit

Thermodynamical limit

XE(S)eµn(S) estates S

Inverse temperature
Chemical poten5al
Energy=1kB TnD ! +1
l !0

Discrete Einstein-Hilbert par55on func5onXe D nD (ST ) eD 2 nD 2 (ST )STconnectedtriangulation

Inverse of Newton constantµD ?

E(S)nD 2 ?

D 2 /1G

Not bounded from below…

14

Quantum gravity? Analogy with thermodynamics…
Microscopic descrip4on of gaz

Microscopic descrip4on of space-4me

Accessible state S

Triangula5on STn(S) par5clesnD (ST ) D-simplices

n ! +1Dist ! 0
Grand-canonical par55on func5onCon5nuum limit

Thermodynamical limit

XE(S)eµn(S) estates S

Inverse temperature
Chemical poten5alEnergy=1kB T

nD ! +1
l !0

Discrete Einstein-Hilbert par55on func5onXe(aD 2 D )nD e D 2 (anD nD 2 )STconnected
triangulation

Inverse of Newton constantD 2 /µaD 2DE(S)anDnD 21G15

a ??

2 – Discrete quantum gravity
Discrete Einstein-Hilbert par55on func5on=e
N =eaD2

D

F( , N) == e D 2X

connected
triangulations

1CnD

N(anD nD2)

① Large N limit (Physical limit of small Newton constant) :a has to be chosen such that anD

nD 2 bounded from below

16

2 – Discrete quantum gravity
Discrete Einstein-Hilbert par55on func5on=e
N =eaD2

DF( , N) == e D 2X

connectedtriangulations

1CnDN(anD nD2)

① Large N limit (Physical limit of small Newton constant) :a has to be chosen such that anDnD 2 bounded from below

à well-deﬁned 1/N-expansion:

11
1
F ( , N ) = F0 ( ) + F1 ( ) + 2 F2 ( ) + · · ·NkN
N(k = D?)

“Genera5ng func5on” of connected triangula5ons thatnD 2minimize anD

16

2 – Discrete quantum gravity
Discrete Einstein-Hilbert par55on func5on=e
N =eaD2

DF( , N) == e D 2

X

connectedtriangulations

1CnDN(anD nD2)

① Large N limit (Physical limit of small Newton constant) :a has to be chosen such that anDnD 2 bounded from below② Con5nuum limit l ! 0 à should be non-trivial

a has to be chosen such that inﬁnitely many spaces minimize anDnD 217

2 – Discrete quantum gravity
Discrete Einstein-Hilbert par55on func5on=e
N =eaD2

DF( , N) == e D 2Xconnected
triangulations

1CnD

N(anD nD2)

① Large N limit (Physical limit of small Newton constant) :a has to be chosen such that anD

nD 2 bounded from below② Con5nuum limit l ! 0 à should be non-trivial

a has to be chosen such that inﬁnitely many spaces minimize anDnD 2

Con5nuum limit <-> singularity of F0 à asympto5cs of F0 (string suscep5bility…)
17

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nD 2nD%""""!"-./0123%4567089:71;0<%=>6?>%@7A6@6B/%%%%%%%%%%%%%%%%74%CA/.%nD
Im"%-D$%$Z(".<(");/?-.;/("-."[D(0""""""""""m""%$,$%$Z(".<("(,(/12J"""48".<(2"+(<-?("-# an"D " nD " 2 = """"""""5".<(,"@(")-,")<**#("
k.<$#"""""=""

a!! D09@/574/%4>/%?;55/<E;0.608%4567089:71;0<%%I.*"*+.-$,".<("'-/1("U")*//('-7*,"8;,)7*,#J""""•! C<-."0*"@("/()*?(/"$,".<(")*,7,;;%"'$%$.B"!! "F5;E/51/<%;2%:758/%<9?>%4567089:71;0<G%
C<-."-/(".<("_-;#0*/n"0$%(,#$*,5"8/-).-'"0$%(,#$*,…B"3b"Y*/("1(,(/-'"#(o,1S"1';("-,2"6$,0"*8"+;$'0$,1"+'*)6#%•! A-,"@("[,0""""""#-7#82$,1")*,0$7*,#""""""""-,0"""""""B"a•! C<-."0*"@("/()*?(/"$,".<("'-/1("'"'$%$.B""nD 2nD%""""!"-./0123%4567089:71;0<%=>6?>%@7A6@6B/%%%%%%%%%%%%%%%%74%CA/.%nD
Im"%-D$%$Z(".<(");/?-.;/("-."[D(0""""""""""m""%$,$%$Z(".<("(,(/12J"""48".<(2"+(<-?("-# an"D " nD " 2 = """"""""5".<(,"@(")-,")<**#("
k.<$#"""""=""a!! D09@/574/%4>/%?;55/<E;0.608%4567089:71;0<%%I.*"*+.-$,".<("'-/1("U")*//('-7*,"8;,)7*,#J""""!%H>/%?;@I6074;567:%
E5;I:/@%;2%4>/%6045;.9?1;0JJJ%•! C<-."0*"@("/()*?(/"$,".<(")*,7,;;%"'$%$.B"!! %F5;E/51/<%;2%:758/%<9?>%4567089:71;0<G%
C<-."-/(".<("_-;#0*/n"0$%(,#$*,5"8/-).-'"0$%(,#$*,…B"3b"3 – Previously known results: D=2

3 – Previously known results: D=2
In dimension D=2, gluings of polygons are combinatorial maps (=ribbon graphs)
Combinatorial maps are discrete D=2 surfaces
The curvature of a map only depends on its genus!

Discrete sphere(Genus = number of holes)

Discrete torusà The enumera5on of maps of a given genus is a very ac5ve domain of research
since the 60’s (Tufe, Bender, Canﬁeld, and so many more …)

19

3 – Previously known results: D=2
à

a = 1/2

à Large N limit selects all discrete spheres

à Con5nuum limit is the Brownian sphere,
A random con5nuous metric space with Hausdorﬀ
dimension 4 (Marckert, LeGall, Miermont… 2006…)

Equivalent to Liouville D=2 quantum Gravity(Conjectured by physicists in 80’s, 90’s…
…math. proof by Miller & Sheﬃeld 2016…)

Fig : J. Bernelli20

3 – Previously known results: colored triangula5ons in any D

K"–"L/(?$*;#'2"6,*@,"/(#;'.#S"./$-,1;'-7*,#"$,"-,2"G"!!"" a = D(D"

1)/4

"

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I…!'0*;#…"3ccf…>;/-;"q"P2-,"Ff3MJ"""%P!6<7EE;60108%:6@64%25;@%4>/%8/;@/456?%%
E;604%;2%M9/%P%%F3"

4 – Some recent results

4 – Some recent results
Dimension 3

Octahedra à Branched polymers (Bonzom, L.L. 2016)

a = 11/8

Simplest torus à “Branched polymers” (Bonzom, L.L., Rivasseau 2015, L.L., Thürigen 2017) a = 1
All colored-triangulated spheres (balls) à “Branched polymers” (Bonzom 2018)(a is known)…To be con1nued, but quite disappoin1ng…

22

4 – Some recent results
Dimension 4
Building blocks of size 4 à 3 cri4cal regimes (Bonzom, Delepouve, Rivasseau 2015) a = 3/2
Building blocks of size 6 à same 3 cri4cal regimes (L.L., Thürigen 2017)“Branched polymers”(= con1nuum tree)“Prolifera5on of baby
universes”= 1/3à Cac1 of BS ?

(a is known)“2D quantum gravity”(= Brownian sphere BS)
23à In D=2, the cri5cal behavior of large N surfaces does not depend on thediscre5za5on of the boundary, it is universal (2D quantum gravity).

à In D=3, as far as we know, it also seems universal (branched polymers).

à In D=4, the cri5cal behavior of maximal curvature
conﬁgura5ons is NOT universal…
… it depends on the building block… Need to keep exploring!

24

Can we ﬁnd new cri5cal regimes this way??
Can we ﬁnd suitable Brownian con5nuum volumes some other way?
(see ongoing work in the last slides)25

5 – Random tensor models

5 – Random tensor models
Introduced in early 90’s by: Ambjorn et al, Sasakura, Gross
1/N expansion and melonic graphs in 2010-13: Gurau, Rivasseau, Bonzom, Riello, Ryan …
Recent developments presented in this talk: Rivasseau, Bonzom, Delepouve, L.L., Thürigen

Matrix models
Interac4on:

M
M̄

M̄

M

M

Tr (M M † )3M̄Par44on func4on:

Z=Z



= e

⇥

e N Tr M M†@1@N Tr @M @M †† 3(M M )e⇤dM dM †N Tr(M M † )3
M =0
M
M̄

M

M̄

à Gluings of hexagons!

MMM̄
M̄

M

M̄
M

M̄

26

Matrix models
1/N expansion of 2 point func4on:

Gp (N, ) =X
g 0

N 2 2g Gp,g ( )where the Gp,g are genera5ng func5ons of connected rooted gluings of p-gons of genus g

Gp,g ( ) =X

cp,g,n nn 0

the coeﬃcients cp,g,n being the number of rooted surfaces of genus g made of n p-gons

à Use matrix models to count surfaces!
A few names (among so many more): ’t Hooz, Kazakov, David, Itzykson, Zuber, Ginsparg,
Di Francesco, Guiter, Bourer, Eynard…

27

5 – Random tensor models
Interac4on:
Invariant under U (N )D ) Speciﬁc colored structureT̄

Trocta (T, T̄ )Par44on func4on:

ZB =Ze N

D1⇥

T.T̄

s

N TrB (T,T̄ )

T̄
T

⇤

TTT̄

dT dT̄
Find the correct value!!
(at most a unique ra5onal…L.L. 2018)

TT̄

T̄

T

TT̄

28

5 – Random tensor models
1/N expansion of 2 point func4on:

GB ( , N ) =XnD

N nD 2 anD

G2G(B)
connected
rootedGluings of building blocks B

Where:a = (D1)

⇣ |B|21⌘sFind the right s

Find the right a…see the discrete QG discussion in early slides31

5 – Random tensor models
1/N expansion of 2 point func4on:

GB ( , N ) =XnD

N nD 2 anD

G2G(B)
connected
rootedGluings of building blocks B

Where:a = (D1)

⇣ |B|
21

⌘

s

Find the right s

Find the right a…see the discrete QG discussion in early slides

If s is well chosen, we have a well-deﬁned 1/N expansion, with inﬁnitely many terms per(non-empty) order.
As for matrix models, the tensor models count gluings of building blocks, according to somewell chosen generaliza5on of the genus.
31à The conclusions from last sec5on also apply for tensor models!

Conclusions

Conclusions§ Recent exact results iden5fy 3 universality classes for some simple Euclidean
dynamical triangula5ons / random tensor models
à escaped the branched polymer phase in DTà new classes from Euclidean DT? Not excluded but s5ll open ques5on§ We can iden5fy and count exactly the large N spaces for many building blocks in
D=3, very few in D=4 + con5nuum limit (?)
We can iden5fy and count the large spaces contribu5ng at any order in 1/N for
triangula5ons and for a few others (=double scaling)§ These combinatorial techniques apply to the iden5ﬁca5on of graphs contribu5ng
to the SYK model (and SYK-like tensor models), for which we can also iden5fy thegraphs contribu5ng at any order…
34

Conclusions
Very diﬀerent needs!!

Discrete QG models:Need higher dimensional random geometry
to emerge at large N in the con5nuum(NOT branched polymers)… Currently unknown (and hard to ﬁnd…)

Large N graphs

SYK-like models:Need solvability at large N
à Tree-like graphs
à Branched polymers in the con5nuum… The large majority of models!!
35

Some ongoing work
1 – Random tensor models.
(e.g… Description at any order for all the tree-like theories (SYK-like). [with S. Dartois])
2 –New continuum limits (Brownian volumes) from more direct approaches?
[one project with S. Dartois, another one with JF. Marckert]

3 – Enumeration and statistical properties of graphs contributing to the (colored) SYKmodel at any order in 1/N [with E. Fusy & A. Tanasa]
4 – Methods apply to ``quantum information’’ problems (probability that a multipartitestate is entangled) [with S. Dartois & I. Nechita 1808.08554]
5 – Non-linear differential equations involved in turbulences, with random initialconditions and coefficients [with S. Dartois & V. Rivasseau & O. Evnin & G. Valette]
5 – Study of the properties of the wave function of the canonical tensor model[with N. Sasakura]

36

November 5-9 // Nagoya UniversityOrganization with Yuki Satohttps://discrete-nagoya.sciencesconf.org

November 5-9 // Nagoya University
Organization with Yuki Satohttps://discrete-nagoya.sciencesconf.org

Thank you for yourattention!