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181 Page 4 of 19 Eur. Phys. J. C (2016) 76 :181
2(dlnX
2
(σ )/dlnσ)
−1
, where X
2
(σ )=
d
D−1
xx
2
P(x, 0,σ). For a set with constant walk dimension,
X
2
(σ )∼σ
2/d
w
. (6)
In a space with walk dimension d
w
= 2 in the IR and d
w
>
2 in the UV, the ratio (d
X
2
/dσ)/(
X
2
/σ ) ∼ 1/d
w
between the differential velocity of the particle (measuring its
local random motion) and the total finite-difference velocity
decreases in the UV. This means that the trajectory of the
probe becomes less ragged than usual in the UV; a pictorial
demonstration of such behavior can be found in [37].
2.1.2 Dimensions of discrete and combinatorial structures
In several quantum-gravity approaches, there is no funda-
mental continuous spacetime. Nevertheless, it is possible
to generalize the above operational definitions to a discrete
set and to extract, only in certain regimes, sensible multi-
scale profiles for the Hausdorff, spectral and walk dimen-
sion. A proof of concept is given in [4,10,11,25–28]. For
instance, a generalization of discrete exterior calculus [52–
54] allows one to construct Laplacians on combinatorial
structures and hence define diffusion processes thereon. The
spectral dimension as well as d
w
and d
h
can be computed for a
class of quantum-gravity states, in particular those appearing
in loop quantum gravity, spin foams, and group field theory
[10,11].
The main requirement we ask in order to have dimen-
sional flow in discrete (pre-)geometries is that there exist
regimes where all three geometric indicators d
h
, d
s
, and d
w
are real-valued and positive. These regimes should extend
from the infrared down to some effective UV scale below
which discreteness or quantum effects destroy some or all of
the indicators, for instance if the expectation values on the
chosen states become complex. In general, the effective UV
scale is determined by the choice of states. The existence of
a regime where quantum geometry has well-defined dimen-
sions translates into regularity assumptions on the quantum
states [11]. Beyond these assumptions, one can plunge into a
wild jungle of quantum-geometry configurations with prop-
erties completely different from classical or semi-classical
spacetimes.
2.1.3 Dimensions of spacetimes
In a continuous spacetime with D topological dimensions,
the definition of Hausdorff dimension is unchanged but
for the addition of the Euclideanized time direction. Then
the Hausdorff dimension of spacetime is the scaling of
the volume of the D-ball. For the spectral dimension, one
includes (imaginary) time in the operator K(∇), while in the
walk dimension time is included in Euclideanized distances
X
2
(σ ) = T
2
(σ )+X
2
1
(σ )+···. Similar considerations apply
to discrete geometric or pre-geometric structures.
In the extension of all the above definitions, one takes the
time direction in Euclidean signature. This step is fairly stan-
dard when one wants to define the dimension of a geometry
with Lorentzian signature. If, for any reason, one cannot or
does not want to Euclideanize time, then it is necessary to
consider the dimensionality of spatial slices and the time line
separately, instead of the whole spacetime. The reader may
adopt whichever point of view they might prefer; this does
not affect the following.
The spectral dimension is sometimes regarded as a theo-
retical parameter useful to classify spacetimes but that does
not correspond to a physical observable. Elsewhere [8,20],
the author and collaborators had already the occasion to
advance a different view: the spectral dimension should be
a meaningful observable just like the topological and Haus-
dorff dimension are. In that case, however, its definition must
be well posed at all stages to make sense physically: if we
want d
s
to be a physical observable (our working hypoth-
esis here), its definition must provide also an operational
way to measure it. For instance, how can we interpret the
parameter σ if time is in the operator K? Also, in certain
cases the form of K is such that P(x, x
,σ) is not a proba-
bility and there is no well-defined underlying diffusion pro-
cess at all (this is a well-known problem in transport theory
with higher-order or non-local operators [42] and in quantum
gravity [8,20]). Mathematically there is no issue whatsoever.
If σ = t , one can enact a fictitious diffusion process with
some Monte-Carlo time on the geometric or pre-geometric
structure one wants to explore, let it be a continuous mani-
fold or the graph ensembles of discretized gravity. Even when
P is not positive semi-definite and the picture of a diffusing
probe fails, to determine d
s
one only needs to consider closed
paths and integrate over them with a certain measure. How-
ever, this is insufficient to characterize an operational way to
physically measure d
s
. In two different interpretations of the
spectral dimension, valid in any regime where an effective
field theory can be formulated, the diffusion equation is a
renormalization-group running equations depending on the
IR cut-off scale k = f (σ ) [8] or, alternatively, it stems from
the Schwinger representation of the particle propagator and
√
κσ = is a length scale determining the resolution 1/ at
which the geometry is probed [20]. The interpretation of the
parameter σ is unimportant in the following but it is worth
to mention these caveats anyway.
2.2 Fine structure and dimensional flow
The first property of the list 1–6 is that a (multi-)fractal set
F should have “a fine structure.” By this, one means that it
is possible to find points of F at all scales of observation,
no matter how deeply one zooms into the set. For contin-
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