Random geometry and the Kardar-Parisi-Zhang universality class

We consider a model of a quenched disordered geometry in which a random metric is defined on ${\mathbb R}^2$, which is flat on average and presents short-range correlations. We focus on the statistical properties of balls and geodesics, i.e., circles and straight lines. We show numerically that the roughness of a ball of radius $R$ scales as $R^\chi$, with a fluctuation exponent $\chi \simeq 1/3$, while the lateral spread of the minimizing geodesic between two points at a distance $L$ grows as $L^\xi$, with wandering exponent value $\xi\simeq 2/3$. Results on related first-passage percolation (FPP) problems lead us to postulate that the statistics of balls in these random metrics belong to the Kardar-Parisi-Zhang (KPZ) universality class of surface kinetic roughening, with $\xi$ and $\chi$ relating to critical exponents characterizing a corresponding interface growth process. Moreover, we check that the one-point and two-point correlators converge to the behavior expected for the Airy-2 process characterized by the Tracy-Widom (TW) probability distribution function of the largest eigenvalue of large random matrices in the Gaussian unitary ensemble (GUE). Nevertheless extreme-value statistics of ball coordinates are given by the TW distribution associated with random matrices in the Gaussian orthogonal ensemble. Furthermore, we also find TW-GUE statistics with good accuracy in arrival times.


Introduction
Stochastic geometry is a branch of mathematics [1] with deep connections to physics, ranging from statistical mechanics to quantum gravity [2,3]. For example, thermal fluctuations of important biophysical objects, like fluid membranes, can be naturally accounted for through the framework of random geometry [4,5]. The effect of thermal or quantum fluctuations of the geometry on systems featuring strong correlations, such as those underlying a continuous phase transition, is typically relevant, in the sense that they modify the values of the critical exponents [6]. For 2D systems, this modification is governed by the celebrated Knizhnik-Polyakov-Zamolodchikov equations [7]. If, instead of thermal or quantum fluctuations, we consider quenched disorder in the geometry, one is naturally led to the study of models like first passage percolation (FPP) [8,9]. In this discrete model, each link of a regular lattice is endowed with a random passage time. FPP theory studies the probability distribution of traveling times between pairs of lattice points. Alternatively, traveling times can be regarded as distances, thereby defining a random metric. Being a generalization of the Eden model [10,9], FPP has played an important role in statistical physics, as an important step for the analysis of other interacting particle systems like the contact process or the voter model. In recent times, additional interest on the model derives from its properties when defined on realistic (disordered) graphs [11], such as those occurring in e.g. communications or economic systems [12].
Inspired by studies in FPP, recent works have dealt with geodesics and balls in a two-dimensional plane endowed with suitable random metrics [13,14]. By suitable, we mean that the metric is on average flat and presents only short-range correlations. In other terms, the geometric properties are considered over distances much larger than either the curvature radius or the correlation lengths. The geodesics on these random manifolds present many interesting properties. Let us consider two points which are separated by an Euclidean distance L. The minimizing geodesic on the random metric which joins them can be regarded as a random curve, when viewed from the Euclidean point of view. Its maximal deviation from the Euclidean straight line grows as L ξ , where ξ = 2/3 [13,14]. It is also possible to study balls on these random metrics. The ball of radius R around any point will be also a random curve, from the Euclidean point of view. For large R, the shape of this curve can be shown to approach a circumference, whose radius is proportional to R. It lies within an annulus whose width grows as R χ , with χ = 1/3 [13,14].
The so-called wandering and fluctuation exponents, ξ = 2/3 and χ = 1/3, for the geodesic and ball fluctuations, respectively, denote a certain universal fractal nature of straight lines and circles on a random geometry. Actually, they are analogous to similar exponents occurring in FPP on a lattice, for which such specific values are known to correspond, through an appropriate interpretation [15], to those characterizing the dynamics of a growing interface. Basically, the boundary of FPP balls can be thought of as such an interface which, in the wider context of growth models [16,17], is expected to evolve subject to an irreversible growth trend, competing with time-dependent fluctuations and smoothing mechanisms. Starting with a flat or a circular form, the interface roughness (mean-square deviation around the mean interface position) grows in time as W (t) ∼ t β . Also, the interface fluctuations present a lateral correlation length which grows with time as ℓ(t) ∼ t 1/z . The FPP values for the growth and dynamic exponents, β = 1/3 and z = 3/2, respectively [15], correspond to those of the so-called Kardar-Parisi-Zhang (KPZ) universality class for one-dimensional interfaces [16,17,18]. Actually, a landmark scaling relation that holds among exponents for systems within this class, namely the so-called Galilean relation β + 1 = 2/z, implies through the interface interpretation [15] that χ + 1 = 2ξ, which has been proved only very recently for FPP under strong hypothesis [19,20]. Rigorously speaking, the individual values of ξ and χ remain conjectural for FPP.
In recent years, evidence has gathered, showing that systems in the KPZ universality class do not only share the values of the scaling exponents β and 1/z, but also the full probability distribution of the interface fluctuations [21]. This applies to discrete models [22,23,24], experimental systems [25,21,26,27], and to the KPZ equation itself [28,29,30]. For one-dimensional interfaces and within the context of simple-exclusion processes -and as a confirmation of a conjecture formulated on the context of the polynuclear growth model [31,32]-it has been rigorously proved that, for a band geometry, interface fluctuations follow the Tracy-Widom (TW) probability distribution function associated with the Gaussian orthogonal ensemble (GOE), while for a circular setting they follow the TW distribution associated with the Gaussian unitary ensemble (GUE) [33,34,35]. Universal fluctuations of TW type are also known to show up in FPP systems, but in those cases the variable whose fluctuations are typically considered is the time of arrival, rather than the radius [36].
Back to random metric geometry [13,14], the values of the fluctuation and wandering exponents suggest a direct relation to non-equilibrium processes in the KPZ universality class. In this work we develop an adaptive numerical algorithm to explore the shapes of balls in arbitrary two-dimensional Riemannian manifolds, and specialize it to work on random metrics of the desired properties. Our algorithm is based on the one used to solve the covariant KPZ equation [37,38]. We show numerically that those balls, as conjectured, follow KPZ scaling. Minimizing geodesics are studied, and their fluctuations are shown to scale in the expected way. However, radial fluctuations are shown not to follow TW statistics. This apparently negative result is explained by studying the fluctuations of a more suitable observable, namely, the time of arrival.
To this end, we perform numerical simulations of FPP on the square lattice in which we explicitly assess the analogy between the latter and our random geometry system. This paper is structured as follows. Section 2 discusses the basics of geometry in random metrics. The numerical algorithm is described in section 3, followed by a detailed study of balls and geodesics in section 4. The failure of radial fluctuations to obey TW statistics leads us to a careful study of FPP in section 5. This hints to the study of the time of arrival in the original random metrics system, which is performed in section 6. Section 7 ends by presenting our conclusions.

Geometry in random metrics
Let us consider the Euclidean plane R 2 , endowed with the usual Euclidean distance, d E . Let us now define a manifold M obtained when a (smooth enough) metric tensor field g is imposed upon R 2 , inducing a distance function d g . The metric tensor field g can be considered as a field of symmetric matrices. Let us consider, following [13,14], an ensemble of random metric fields that fulfills these conditions: • The metric tensor field is as smooth as required.
• Metric tensors at points further away than a given cutoff r 0 are considered as statistically independent. • The distribution of the metric eigenvalues is translationally invariant.
The metric tensor field g can be visualized as a mapping that attaches to each point two orthogonal directions, v 1 and v 2 , and two metric eigenvalues, λ 1 and λ 2 . Alternatively, we can think that each point in R 2 gets an ellipse attached, with principal directions v 1 and v 2 , and semi-axes λ 1 and λ 2 . The geometrical meaning of this ellipse is the following: a particle moving away from the point at unit speed in the manifold M would move in R 2 with a speed given by the intersection of the ellipse with the ray which the particle follows. Now let us choose a point X 0 (e.g., the origin) and consider the set of points, B X0 (r) = {X | d g (X, X 0 ) ≤ r}, whose distance to it is smaller than or equal to a certain r. Since X 0 will remain fixed from the beginning, we will usually drop the subindex. This ball need not be topologically equivalent to an Euclidean ball, since it need not be simply connected. Therefore, its boundary ∂B X0 (r) will consist of a certain number of components, see figure 1 for a pictorial image.
The results in [13,14] guarantee that when this boundary, ∂B X0 (r), is viewed from the Euclidean viewpoint, it lies within two circles centered at X 0 , whose radii scale linearly with r. It is not hard to prove that one of the components of the ball boundary encloses all the others, namely, the one whose interior contains X 0 . Thus, the ball-boundary consists of an outer irregular front plus an internal froth, or set of bubbles. Let ∂ 0 B X0 (r) denote this exterior component.
x 0 Figure 1. A typical ball B X 0 (r) of radius r around a given point X 0 (shaded area). In general, B X 0 (r) need not be simply connected. Therefore, its boundary ∂B X 0 (r) (solid lines) contains different components. The boundary is enclosed between two circles (dashed lines), whose radii grow linearly with r [13,14].
A useful mental image of the ball is a swarm of particles emanating from X 0 , each one escaping from there with unit speed and following a geodesic line. At time t, the set of visited points will be B X0 (t). In this way, the bubbles can be considered as "hills" which are hard to climb. This picture can be made more precise in the following way. Let us consider the tangent space at X 0 , T X0 , and the set of unit vectors, u θ , parameterized by some angle θ. Each u θ determines a unique geodesic curve, γ θ . If each geodesic is traversed at unit speed, then time is a natural (arc-length) parameter: The equality does not generally hold, since many geodesics are non-minimizing [39].
In fact, {θ, t} constitute a -possibly degenerate-coordinate system on the manifold that generalizes polar coordinates. It has a very interesting property: lines of constant θ and lines of constant t are always g-orthogonal. Building from this assertion, one can state a modified Huygens principle for the propagation of the ball front. Given the front at a certain time ∂B(t), it is possible to obtain the front at t+δt, ∂B(t + δt) by allowing each point X on it to move along the local normal direction X → X + δt · n. This is, of course, in analogy to the original Huygens principle for the propagation of light, or Hamilton-Jacobi equation in mechanics.
Let us start with an infinitesimal circle centered at X 0 . Then the ball for time t fulfills simply the equation where X stands for a generic point on ∂B(t) and n g (X) is the local normal to such an interface, with respect to the metric g. We can gain some intuition about this Huygens principle from figure 2, which shows a zoom on a region of the ball front. The dashed ellipse shows the local metric tensor g. How to obtain the normal vector n g , given the tangent t and the metric? The g-orthogonality relation t ⊥ g n g can be stated as g µν t µ n ν g = 0, i.e.: g t ⊥ n g , where ⊥ denotes the Euclidean orthogonality relation. Application of g to t makes it always closer to the principal direction with maximal eigenvalue. Therefore, n g will always be closer to the principal direction with minimal eigenvalue. Of course, n g must be g-normalized, so that the front will move with unit speed in M.
Therefore, if the metric is given, the propagation algorithm can be summarized as follows: t ∂B(r) g t n g Figure 2. A small region of the ball front, showing the local tangent ( t) and normal ( ng) vectors. They are orthogonal only with respect to the metric g, which is represented with the dashed ellipse. Notice that t and ng are not orthogonal within the Euclidean framework. Instead, in the Euclidean metric ng is orthogonal to g t, which is the correct notion of g-orthogonality.
• For each point of the front, find t.
• Compute g t.
• Find an Euclidean normal to that vector, N g . Of course, take good care of the orientation! • Normalize that vector according to g, namely, find | N | 2 g ≡ g µν N µ g N ν g and compute • Move the point by the vector quantity δt · n g .

Numerical simulation algorithm
We have adapted our intrinsic-geometry algorithm for the covariant KPZ equation, employed in [37,38], to the simulation of the balls in generic metrics. In our approach, we simulate the ball propagation of equation (1), starting out with an infinitesimal circle, and allowing time to play the role of the ball radius. The ball at any time will be given by a list of points in the plane. The spatial resolution of the front is held constant: the Euclidean distance between two neighboring points ∆x must stay within a certain interval [l 0 , l 1 ]. This is achieved by inserting or removing points in a dynamical way. Moreover, self-intersections can appear naturally, as anticipated in figure 1. In such cases, we retain only the component which contains the origin of the ball, i.e., we track ∂ 0 B(r).
For illustration, figure 3 shows the integration procedure as applied to several deterministic metrics. In each case, the metric g is obtained from the first fundamental form of a simple surface. Ideed, the form of the corresponding balls in the Euclidean plane intuitively reflect the "speed" with which the interface (ball) grows at each point as a function of the value of the metric there.
Generation of a random metric tensor field is performed by assuming that the correlation length is shorter than the cutoff distance assumed for the ball, i.e., r 0 < l 0 . Thus, the metric tensors at sampled points are statistically independent. The procedure does not require derivatives of the metric tensors, as it would if one insisted on tracking individual geodesics. The metric tensor at each point is specified by providing the two orthogonal unitary eigenvectors, v 1 and v 2 , and the two corresponding eigenvalues. Thus, v 1 is generated randomly, v 2 is just chosen to be leftwards orthogonal to it, and the eigenvalues are uniform deviates in the interval [λ 0 , λ 1 ], where λ 0 should be strictly larger than zero. The balls are analyzed from the Euclidean point of view: their roughness W is found after fitting to an Euclidean circle, and by computing the average squared deviation from the ball points to it, ultimately averaging over disorder realizations. We will also consider the standard deviation σ r of the radius of the fitting Euclidean circle over realizations of the disorder. Thus, W can be interpreted to quantify intra-sample radial fluctuations, while σ r assesses inter-sample radial fluctuations.

Balls and geodesics in random metrics
The algorithm described in the previous section has been applied to integrate equation (1) numerically for different realizations of the disorder in the geometry. Our simulations start with a very small ball, with initial radius 0.05, and propagate it through a random metric with eigenvalues λ ∈ [1/20, 1]. The time-step used is ∆t = 5 · 10 −3 and the ultraviolet cutoff for the simulation is chosen to be [l 0 , l 1 ] = [0.01, 0.05]. Results were checked to remain unchanged for smaller values of the discretization parameters. Figure 4 shows an example of balls with increasing radii for times (i.e., g-radii) in the range t = 0.2 to 3.4.

Roughness and radial fluctuations
We have simulated equation (1) for 1280 realizations of the disorder in the metric and analyzed the Euclidean roughness of the resulting balls as a function of time (i.e., g-radius). The results for the roughness W (t) are shown in figure 5. The figure also shows the evolution of the standard deviation of the average fitting Euclidean radius, σ r , for each time. Both observables are shown to follow power-law behavior: W ∼ t χ , with χ ≃ 1/3 (0.3326 ± 0.0001). The standard deviation of the Euclidean radii, σ r , also scales as a power law, but with a different exponent, σ r ∼ tχ, withχ = 0.185 ± 0.001.
The relation of the χ exponent of random geometry and the β of the KPZ universality class is evident. In both cases, the interface roughness grows as a powerlaw of time. There is, notwithstanding, a difference in the behavior of the standard deviation of the radii. Other studies have shown [38] that, within the KPZ universality class, σ r follows a power-law behavior with the same exponent β as the roughness. This equality between the inter-sample and intra-sample radial fluctuations is linked to the KPZ class, and does not necessarily follow in other growth regimes.

Geodesic fluctuations
Our next numerical experiment consists in obtaining the average lateral deviation of the minimizing geodesics. We define this geodesic fluctuation in the following way. Consider two points which are an Euclidean distance L apart. Find the minimizing geodesic joining them, and mark also its middle point M , see figure 6. The Euclidean distance from this point to the straight line joining both points is δ ⊥ , the lateral fluctuation of the geodesic. Moreover, we can find the Euclidean distance from M to the perpendicular bisector of the aforementioned straight line, δ , namely, the longitudinal fluctuation of the geodesic. Notice that the disorder averages of both δ and δ ⊥ should be zero, but their root mean square is highly informative. According to previous work [13,14], they should scale as δ ∼ L χ , with the same exponent χ = 1/3, and δ ⊥ ∼ L ξ , with ξ = 2/3, a new critical exponent.
We have proceeded to estimate both δ ⊥ and δ , i.e., the lateral and longitudinal fluctuations of the geodesics, between points A = (−L/2, 0) and B = (L/2, 0), for different values of L and 128 realizations of the disorder. We have proceeded as follows: We simultaneously grow two balls centered at these points. Growth is arrested when both balls intersect for the first time. The coordinates of their first intersection point are, precisely, M = (δ , δ ⊥ ). See figure 7 (left) for an illustration. The rationale is as follows. Let us call t x the time (g-radius) at which both balls first intersect. Point M can be reached in time t x both from A and from B, hence it should be on the minimizing geodesic connecting both points.
In order to save simulation time, each simulation is carried out in practice as follows: we start with two very small balls separated by a small distance L 0 , and grow them until they first intersect. At this moment, we take note of the coordinates of the intersection point, increase the separation of the balls by ∆L, rotate each one by a random angle, and continue the simulation until they intersect again. This procedure is repeated until the desired range of L has been covered. The random rotation ensures that the ensuing intersection points are uncorrelated.  We obtain the root-mean-square horizontal and vertical deviation of the intersection point as a function of the Euclidean distance between the two points. The results appear in figure 7. The lateral fluctuations of the geodesics scale with the separation L between the ball centers as δ ⊥ ∼ L ξ , with ξ ≃ 2/3, while the longitudinal fluctuations scale with the same exponent value as the roughness, namely, δ ∼ L χ , with χ ≃ 1/3. It is straightforward to understand the exponent for the longitudinal fluctuations, as δ is quite naturally expected to grow with the ball roughness. Through the rough interface interpretation mentioned above [15], the lateral fluctuations δ ⊥ , are otherwise related to the increase in the correlation length characteristic of systems in the KPZ universality class, namely, δ ⊥ (t) ∼ ℓ(t) ∼ t 1/z , with 1/z = 2/3. Indeed, the exponent values we obtain for the random metrics system are compatible, within statistical uncertainties, with the so-called Galilean relation, χ+1 = 2ξ, characteristic of the KPZ universality class. The geometrical interpretation of this exponent identity within such context is as the expression, within a scaling hypothesis, of the fact that on average the rough interface grows with uniform speed along the local normal direction [15], implementing a Huygens principle as discussed above.

Radial fluctuations
As discussed in the introduction, physical systems for which fluctuations belong to the 1D KPZ universality class are consistenly being found to not only share the values of the critical exponents β and 1/z (respectively, χ and ξ in the random metric language), but also to be endowed with a larger universality trait, alike to a central limit theorem: Radial fluctuations of the interface follow the same probability distribution function as the largest eigenvalues of large random matrices extracted from the Gaussian unitary ensemble (GUE), i.e., the Tracy-Widom GUE (TW-GUE) distribution.  (2), along with the bestfitting Gaussian and TW-GUE distributions. Notice that the TW-GUE fit is not accurate, as the numerical data are much more symmetric. The Gaussian fit has a more correct symmetry, but is inaccurate in the tails, which is evidenced by the discrepance in the kurtosis value. Figure 8 characterizes the probability distribution function for the Euclidean radii of balls of different times (or g-radii). For each time, the ball is fit to a circle with radius r. We collect the (t i , r i ) pairs for 1280 realizations and 171 different times (separated by 0.2 units, up to t max = 35, discarding the first few times). We fit these pairs to a straight line r i = r 0 +vt i . Next, we consider the residuals, ∆r i = r i −r 0 −vt i , and fit their squares to a power-law in time, (∆r i ) 2 ∼ Γ 2 t 2γ . This allows us to define a reduced variable that must have zero mean and unit variance, The Prähofer-Spohn conjecture [31,32] for the radial fluctuations within the KPZ universality class is that γ should be equal to β = 1/3, with the ρ i distribution being time-independent and actually following the TW-GUE pdf. However, as seen above, in our simulations γ ≃ 0.18, while the cumulants of the ρ-distribution are very far away from those of the TW-GUE distribution: The skewness is near zero, and the kurtosis is high, see figure 8. Hence, we conclude that radial fluctuations are by no means TW-GUE-distributed. Does this mean that the balls in random metrics do not fall within the KPZ univerality class, despite the accurate agreement of the critical exponents?

Lessons from first passage percolation
The failure of the radial fluctuations to obey TW-GUE statistics in the random metric system leads us to naturally consider its well known discrete analogue in 2D, namely, first passage percolation (FPP). Let us consider a 2D lattice. Each link k has a certain transit time attached to it, τ k , extracted independently from a certain probability distribution function.
The main question to be asked is about the probability distribution of arrival times t( r) at a given point r. Given the anisotropy of the lattice, it is customary to ask about the probability to reach a point whose separation from the origin corresponds to a lattice segment, e.g., r x = (d, 0). It has been proved that the expectation value of t( r x ) grows linearly with d, and its standard deviation grows as d 1/3 [9]. When properly normalized, and under strong disorder, the fluctuations of the time of arrival do correspond to the TW-GUE pdf [36].
We have tested the extent of the analogy between FPP and random metrics. First of all, we have checked numerically the discrete analogue of the balls in random metrics: their roughness and radial fluctuations. Then, we have carried out explicit simulations of the fluctuations of the times of arrival in FPP. The simulations have been carried out as follows. For a 500 × 500 square lattice, the transit times of all links were extracted from a uniform distribution in [τ 0 , τ 1 ] = [0.1, 5.9], and 12800 samples were obtained. The time of arrival at each lattice site t( r) was computed using a Dijkstra algorithm [40]. The balls were obtained as follows: For a given bound t B , we collect the set, B(t B ) ≡ { r k | t( r k ) ≤ t B }, with sites r k for which the arrival times from the origin equal t B at most, t( r k ) ≤ t B . The boundary of this set, ∂B(t B ), is computed as the subset of sites which have at least one neighbor outside B(t B ). We define the set of balls as ∂B(t) for all t. Figure 9 shows the roughness W of the ∂B(t) balls, and the mean square fluctuations (among samples) in the ball radii, as obtained in our simulations. The roughness grows as a power law, with an exponent higher than expected, W ∼ t 0.41 . But the most remarkable feature of the plot is that the sample-to-sample radial fluctuations scales with an exponent which is significatively smaller, σ r ∼ t 0.22 . The expectation within KPZ scaling is that both magnitudes should scale with the same exponent. Of course, this is not the case. Again, the radial fluctuations do not follow the TW-GUE probability distribution. We next move our focus to the fluctuation of the times of arrival. In a sense, this corresponds to the following duality: • Site dispersion, or roughness refers to the fluctuations of the Euclidean radius of a g-circle. • Time dispersion, or time of arrival fluctuations, refers to the fluctuations of the g-radius of an Euclidean circle. Figure 10 illustrates this duality by showing a 3D plot in which the horizontal plane represents the FPP system. In blue (× symbols), ∂B(t), namely, the ball with g-radius t, is shown for t = 119. Let r be its average Euclidean radius. The vertical coordinate of the red crosses corresponds to the time of arrival at a site on an Euclidean circle of radius r. Note the strong fluctuations in the vertical coordinates.
The cumulants of the fluctuations in the times of arrival are shown in figure 11. Two types of sites are shown: + stands for sites within the main cross, i.e.: with coordinates of the form (x, 0) or (0, y); meanwhile, × denotes sites on the two main diagonals on the plane, (x, x) or (x, −x). By tracking both sets of points we can discriminate which effects are due to the (remaining) anisotropy of the lattice. The mean time of arrival grows linearly with the Euclidean radius, as seen on the upper-left panel of figure 11. The upper-right panel of the figure shows the standard deviation of the time of arrival, indicating the occurrence of power-law scaling with an exponent that, in both cases (+ and ×), is compatible with 1/3. The two lower panels show the skewness and the kurtosis of the distribution of the times of arrival, showing how they slowly approach the TW-GUE values, which are indicated by the horizontal lines. We can conclude that our numerical data agree with the theoretical prediction, namely, the times of arrival follow the TW-GUE distribution for FPP.

Time of arrival in random metrics
Since the random metric system is a continuum analogue of the FPP system, we may expect that times of arrival in the former will grow with Euclidean distance as t(d) ∼ d 1/3 , and that their fluctuations will follow the TW-GUE distribution. As we will show in this section, this is indeed the case. Measuring times of arrival within our scheme requires an special simulation device, illustrated in figure 12. We have scattered a set of checkpoints X i throughout the manifold. At each time step, a winding-number algorithm is performed in order to check whether each one of them is inside or outside the corresponding ball. When point X i changes status from outside to inside, we identify that time as its arrival time. The X i are distributed as a linear golden spiral, i.e., their Euclidean radii increase linearly, but their angles follow the sequence α i = i2πφ, where φ is the golden section, φ = ( √ 5 − 1)/2. This distribution is chosen so as to ensure a uniform angular distribution, as uncorrelated as possible.
The numerical simulations give the expected results, namely, the times of arrival grow linearly with distance to the origin, and their standard deviation also increases with distance, as σ t ∼ d 0.339 , see figure 13 (upper left panel). The higher cumulants are fully compatible with the TW-GUE distribution, giving −0.218 for the skewness and 0.078 for the kurtosis, see upper right panel. The exact values are −0.224 and 0.093, respectively [31,32]). The full histogram, and its comparison with the TW-GUE distribution is shown in figure 13 (lower panel).

Conclusions and further work
We have shown evidence of KPZ behavior in a purely geometric model, where the role of the interface is played by balls of increasing radii in a stochastic manifold, when considered over distance scales which are large compared to either the correlation or  the curvature lengths. When the balls on the random manifold are viewed from an Euclidean point of view, they appear to be rough. If the radius of the balls increases linearly with time, we show that the growth of the Euclidean roughness of the ball is W ∼ t χ , with χ = 1/3. Moreover, study of the minimizing geodesics has shown that the lateral correlations of the fluctuations in the balls scale as ℓ ∼ t ξ with ξ = 2/3. These critical exponent values are the hallmark of the KPZ universality class, although in a different language, namely, χ → β and ξ → 1/z. Our results thus allow to assess numerically the predictions for Riemannian first-passage percolation [13,14], providing a detailed picture of the full stochastic behavior. Given the relation to FPP proper, this detailed description may aid in the development of rigorous proofs that justify completely the values of the wandering and fluctuation exponents in such a discrete model. The relation bears some surprises. First of all, the stochastic geometry model proposed is one with quenched disorder, i.e., the disorder does not change with time. Nonetheless, the resulting universality class is not that of the quenched KPZ equation [16], whose growth exponent is β = 2/3, but rather that of the KPZ equation with time-dependent noise. Given the relation of the model we study with FPP, and in turn the connection of the latter with the Eden model, it is natural to speculate whether our results point towards a detailed relation between the quenched and time-dependent KPZ universality classes. Note that TW fluctuations have been found in paradigmatic systems of quenched disorder, such as spin glasses, structural glasses or the Anderson model [41,42]. This point seems to warrant further study.
The second surprise is related to the absence of the expected Tracy-Widom (TW) fluctuations for the interface radius in the long term regime. Indeed, as opposed to known KPZ models, the deviation of the radial fluctuations σ r does not follow the same scaling as the width of the interface [38]. This result is unexpected and also requires a suitable explanation.
Nonetheless, TW statistics do appear in our model, but in a different setting. Instead of considering the Euclidean fluctuations of the random metric balls, we can consider the random metric fluctuations of the Euclidean balls. In other terms, the time-of-arrival fluctuations of the random balls at different distances from the origin. The deviation of those values does follow the same power-law as the roughness growth in KPZ, σ t ∼ t χ , with χ = 1/3, and the fluctuations follow TW statistics. Due to the wide connections and applications of the FPP model to disordered systems, one can speculate whether this reinterpretation might allow to identify TW statistics in still many other phenomena in which it has not been identified yet. This would strengthen the role of TW fluctuations as a form of a central limit theorem for many far-from-equilibrium phenomena.