Dynamical
Triangulation Approach to Regge Calculus |

**In General**

* __Idea__: A path-integral approach to spacetime geometry in which one sums over piecewise-flat geometries with fixed edge lengths, and varies the triangulation by adding/removing
simplices with well-defined moves.

* __Motivation__: Supposed to be ergodic in the space of geometries, and
one expects to be able to get many more.

* __Drawbacks__: Recovery of
the Einstein-Hilbert action is more problematic, since diffeomorphism invariance
is lost.

* __Action (4D)__: Given by *I* =
2π *N*_{2} – 10 *αN*_{4},
with *α* = arcos(1/5), *N*_{n} =
number of *n*-simplices.

* __2D__: Dynamical triangulations
are equivalent to matrix models.

* __Phase transition__: As
one changes the curvature (Newton) coupling, between an elongated and a crumpled
phase.

**References**

@ __General__: Godfrey & Gross PRD(91)
[more than 2D]; Ambjørn & Jurkiewicz PLB(92);
Nabutovsky & Ben-Av CMP(93)
[4D, non-computability]; Ambjørn
CQG(95)
[2D]; Ambjørn et al LNP(97)ht/96;
Schleich & Witt gq/96-proc
NPPS(98)gq/97.

@ __3D__: Carfora & Marzuoli IJMPA(93)
[and Reidemeister torsion]; Egawa & Tsuda PLB(98)
[random surfaces].

@ __Random surfaces__: David et al NPB(87)
[critical exponents]; Migdal JGP(88).

@ __Related topics__: Renken NPB(97)hl/96 [renormalization
group]; Catterall et
al PLB(98)
[singular geometries]; Henson CQG(09)-a0907 [coarse-graining].

@ __Phase structure, transitions__: Agishtein & Migdal NPB(92),
MPLA(92);
Varsted NPB(94)
[and continuum limit]; Brügmann & Marinari JMP(95)
[no exponential bound]; de Bakker PLB(96)
[phase transition, first-order]; Renken et al NPB(98),
Warner et al PLB(98)
[3D]; Warner & Catterall PLB(00)hl [4D,
with boundary]; Laiho & Coumbe PRL(11)-a1104, Coumbe & Laiho a1201-PoS [Euclidean, asymptotic safety and spectral dimension]; Rindlisbacher & de Forcrand a1311-conf, JHEP-a1503 [4D Euclidean, the phase transition is 1st order].

@ __Euclidean__:
Brügmann PRD(93),
& Marinari
PRL(95) [4D,
measure]; Ambjørn et al JMP(95)ht [2D];
Veselov & Zubkov PLB(04)
[10D]; Ambjørn & Budd APPB(14)-a1310-ln [2D, coupled to matter]; Coumbe & Laiho JHEP(15)-a1401 [non-trivial measure term]; Laiho et al PRD(17)-a1604 [and asymptotic safety]; Laiho et al a1701-proc [recent results].

__Related subjects__: see causal dynamical triangulations; spin-foam models [spincube models].

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send feedback and suggestions to bombelli at olemiss.edu – modified 3 dec
2017