A paper by Zurek applied a random walk argument to a black hole horizon.
Below, we step out our own cosmic de Sitter derivation of the random walk argument. To do this, let $l_{\Lambda}$ represent the generalised de Sitter horizon scale. Due to the holographic UV/IR correspondence, this scale manifests dually: at the fundamental microscopic limit as $l_{UV} = 2L_p$ (the gravitational/casual limit, aka the Schwarzschild radius of a Planck mass), and at the macroscopic cosmological limit as $l_{IR} = c/H$. By mapping between these boundaries, we can derive the limits of the spacetime fluid.
Because spacetime acts as a quantum-critical holographic medium, the microscopic uncertainty accumulates over macroscopic scales, giving us the exact phenomenological signature Zurek found:
$$\Delta x^2 = 2DT$$
In this equation, $\Delta x$ is the position uncertainty, $D$ is the Einstein diffusion coefficient, and $T$ is the time between measurements (the relaxation time).
Einstein Diffusion Coefficient
So, let's look at the Einstein diffusion coefficient.
$\mu = v_d / F$ is the mobility and $v_d$ is the drift velocity. We can write this as
$$\mu = v_d \kappa$$
as the inverse of the Einstein gravitational constant $\kappa$ is a local force, i.e.
$$F = \frac{c^4}{8\pi G} = \frac{1}{\kappa}$$
Drift velocity is acceleration $\times$ relaxation time, i.e. the time between two successive collisions. Using our microscopic UV cutoff $l_{\Lambda} = 2L_p$, if we take the maximal de Sitter acceleration
$$a = \frac{c^2}{l_{\Lambda}}$$
and minimal relaxation time i.e.
$$T = \frac{l_{\Lambda}}{2c}$$
we get
$$v_d = aT = \frac{c^2}{l_{\Lambda}} \cdot \frac{l_{\Lambda}}{2c} = \frac{c}{2}$$
The drift velocity $v_d$ arises because microscopic spacetime fluctuations are locally isotropic, but a causal horizon admits only outward‑directed null steps; the mean outward projection of a fixed‑magnitude null velocity $c$ over the accessible hemisphere is therefore $\langle v_z\rangle = c/2$, making the factor $1/2$ a universal consequence of isotropy combined with one‑sided causal projection, independent of curvature or microscopic details.
Then:
$$D(l_{\Lambda}) = \frac{2cL_{p}^{2}}{l_{\Lambda}}$$ Where $T_{UV}$ is the Unruh temperature associated with the maximal acceleration $a$. At the UV $l_{\Lambda} = 2L_p$:
$$D_{UV} = L_p c$$ This isolates the fundamental microscopic diffusion constant of the spacetime vacuum.
Microscopic Quantum Harmonic Oscillator
At this fundamental scale, over a minimal relaxation time, the position uncertainty of a minimal Quantum Harmonic (QH) oscillator evaluates perfectly to the microscopic limit. Using the zero-point energy $m_s$, which represents the intensive property of the vacuum, the fundamental quantum :
$$m_s = \frac{\hbar}{cl_{\Lambda}}, \quad \omega_B = \frac{c}{l_{\Lambda}}$$
$$\Delta x_{UV} = \sqrt{\frac{\hbar}{2m_s\omega_B}} = \sqrt{\frac{l_{\Lambda}^2}{2}} = \sqrt{2}L_p$$
Macroscopic Horizon Uncertainty
However, in a holographic universe, this microscopic random walk accumulates over the macroscopic observation scale. By evaluating the diffusion over the macroscopic causal horizon ($T = l_{IR}/c$), we obtain a geometric mean, a seesaw relation mapping the UV (Planck) limit to the IR (Cosmological) limit:
$$\Delta x_{IR} = \sqrt{2D_{UV}T_{IR}} = \sqrt{2(L_p c)\cdot\frac{l_{IR}}{c}} = \sqrt{2L_p l_{IR}}$$
$l_{IR}$ is the classical coordinate radius of the universe, $\Delta x_{IR}$ is the quantum thickness of that boundary.
Not so coincidentally, this scale-invariant geometric mean is the exact relation Bousso and Penington (B&P) find for the protrusion distance outside the horizon of an entanglement island from a 4D Schwarzschild black hole.
String-Bit
The Misner-Sharp mass defines the effective active gravitational mass enclosed within a sphere of radius $r$. For our dS CEH the radius is $l_{IR}=c/H$. The CEH mass is an extensive property: \begin{equation}
m_{CEH} = \frac{c^{2} l_{IR}}{2G}
\end{equation} Taking the ratio of the macroscopic mass to the microscopic mass: \begin{equation}
\frac{m_{CEH}}{m_s}
= \frac{l_{IR}}{L_p}
\end{equation}This shows that the CEH mass scales linearly with the 1D radius. This supports not only the 1D diffusion equation we use, but also the 't Hooft-Susskind polymer/string model: the CEH horizon mass scales linearly like a 1D string wrapping the boundary, not a 2D sheet. Mass is then the extensive length of the 1D string. Interestingly, a 1D string self-intersecting on a 2D surface (a space-filling curve) produces a surface code. The fast-scrambling ensures maximum non-locality and error correction. This points vacuum as a maximally error-correcting code, a mechanism for topological protection.
Bekenstein Quantum of Area
If a closed 1D string undergoes a quantum random walk to form a space-filling curve on the 2D spherical boundary of the universe, it will densely interact. If the network is fast-scrambling, we get maximum connectivity where string-bits interact in pairs (nodes). These nodes scale as 2D, and if we assign one unit of holographic entropy to each node rather than string-bit, we get the BH entropy. Entropy is the combinatorial geometry of the 1D string-bit 2D intersections.We can also consider the Bekenstein quantum of area at the microscopic limit:
$$\Delta A = 8\pi L_p^2 = 2\pi l_{\Lambda}^2$$
For the surface area of a sphere with radius $r$, where $r^2 = \Delta x_{UV}^2 = 2L_p^2$:
$$\Delta A = 4\pi r^2 = 8\pi L_p^2$$
Zurek also pointed out that the equality $S_{BH} = S_{ent}$ is known to be true only in certain systems; however, there is evidence this equality holds more generally.
Fluid Dynamics
This derivation invites a direct translation into quantum-critical fluid dynamics. We defined a mobility $\mu$. If we accept the result for $D_{UV}$, we are effectively assigning a fundamental minimum kinematic viscosity to the vacuum spacetime:
$$\nu_{UV} = L_p c$$
This means the Schmidt number of the dS vacuum is one. If the horizon is a fluid, this diffusion calculation shows that it flows with maximum quantum efficiency. Also, the vacuum has no scale because it is at the RG fixed point.
Because we are dealing with a topologically protected quantum-critical fluid, its macroscopic behaviour is governed by the scaled macroscopic kinematic viscosity:
$$\nu_{IR} = \frac{l_{IR}}{2}c = \frac{c^2}{2H}$$
We can evaluate its macroscopic shear viscosity using the macroscopic density $\rho_{\Lambda} = \frac{3H^2}{8\pi G}$:
$$\eta_{vac} = \rho_{\Lambda}\nu_{IR} = \frac{3H^2}{8\pi G}\cdot\frac{c^2}{2H} = \frac{3c^2 H}{16\pi G} = \frac{3\sigma(0)H^3}{16\pi G}$$
where $\sigma(0) = l_{IR}^2$ is a geometric identity, showing a membrane-paradigm / AdS-CFT dictionary. Note $\eta_{vac}$ is not the volume viscosity $\zeta$. Then, as the bulk entropy density $s$ (dimension $k_B L^{-3}$) is:
$$s = \frac{S_{dS}}{V} = \frac{3}{4}\frac{k_B c^3}{\hbar G l_{IR}}$$
we recover the KSS conjecture from the AdS/CFT correspondence (shear viscosity to bulk entropy density ratio $\eta/s \ge \hbar/4\pi k_B$). The reason this works seamlessly across 60 orders of magnitude is UV/IR conformality: the ratio of degrees of freedom to dissipation is topologically protected.
Interestingly, \begin{equation}
\lambda_L \cdot \frac{\eta}{s} = \frac{k_B T dS}{2}
\end{equation} where $\lambda_L=H$ is the MSS bound on chaos and $T_{BH}=T_{dS}/2$. In Planckian fluids, the butterfly velocity $v_B$ is determined by the relationship between the diffusion of energy/momentum and the chaos parameters, with $C$ being a numerical constant of order unity. \begin{equation}
D = C \frac{v_B^2}{\lambda_L}
\end{equation}Here of course, we can easily see that $$ v_B = \frac{c}{\sqrt{2}}$$
Quantum of Circulation
We can also see at the fundamental limit:
$$\nu_{UV} m_s = \frac{\hbar}{2}$$
Onsager noted the ratio of Planck's constant to mass has the same dimension as kinematic viscosity. This implies that the effective kinematic viscosity — aka the quantum of circulation — associated with coarse-grained dynamics can be written as the universal critical velocity $v_c$ associated with oscillatory flow generated by an oscillating body: $$v_c^2 = \nu_{IR} H = \frac{c^2}{2}$$ $$ i.e. $v_B=v_c$. The butterfly is the vortex! In a relativistic conformal fluid the sound speed is $c_s = c/\sqrt{3}$, however, in a holographic universe, information must scramble faster than sound can propagate.
The associated Reynolds Number is $R_e = v_c L / \nu$, where $v_c$ is the critical velocity, $L$ is the characteristic length scale, and $\nu$ is the kinematic viscosity. Contrary to naive expectations, a Reynolds number holds even for superfluids, where for sufficiently rapid flows, perfect inviscid flow breaks down and an effective viscosity emerges dynamically via the nucleation of quantised vortices.Indeed, because this is a quantum-critical fluid, we can evaluate this at any generalised scale $l_{\Lambda}$. Associating the characteristic length scale $L = l_{\Lambda}/2$ and viscosity $\nu = l_{\Lambda}c/2$, the length scales completely vanish:
$$R_e = \frac{v_c L}{\nu} = \frac{(c/\sqrt{2})(l_{\Lambda}/2)}{l_{\Lambda}c/2} = \frac{1}{\sqrt{2}} \approx 0.7$$
The resulting vacuum $R_e$ is a scale-invariant topological constant, identical to that experimentally (Reeves et al, 2014, Identifying a superfluid Reynolds number via dynamical similarity), found as a universal superfluid Reynolds number. Also, turbulence here is not just chaotic geometry, it would be quantised vortex nucleation. Of course, the presence of massive particles breaks conformal invariance (creates drag on the CEH), so this turbulence threshold only applies to pure de Sitter.
Quantum Limit of Diffusion
We can also recover the quantum limit of diffusion as:
$$D \ge \frac{\hbar}{m^*}$$
In our view, where classical GR is simply an effective hydrodynamical description, this connects the Misner–Sharp effective gravitational mass to the CEH mass, i.e. $m^* = m_{CEH}$.
If we consider Schrödinger's equation considered as a diffusion equation, where
$$D = \frac{\hbar}{2m}$$
Now, there is a single scale-dependent mass quantum, $m_s(\ell)$:
\begin{equation}
m_s(\ell) = \frac{\hbar}{c\,\ell},
\end{equation} whose value at the UV (Planck scale) and IR (dS horizon scale) is:
\begin{equation}
m_{s}^{\mathrm{UV}} = \hbar / \left( 2 c L_{p} \right)
\end{equation} \begin{equation}
m_{s}^{\mathrm{IR}} = \hbar H_{\Lambda} / c^{2} = \hbar / (c l_{\mathrm{IR}})
\end{equation} If the macroscopic CEH behaves as a single quantum object, the effective stochastic mass in diffusion is $m_{s}^{\mathrm{IR}}$ rather than $m_{CEH}$. Then:\begin{equation}D_{IR} = \frac{\hbar}{2 m^{IR}_s} = \frac{c^2}{2 H_\Lambda}\end{equation}
This implies that the vacuum acts as a quantum-critical holographic fluid. Also, in GR the horizon is observer-dependent, but the holographic viscosity of spacetime is universal and gauge-invariant. This resolves an apparent paradox: different observers see different geometric horizons, but all measure the same horizon fluid properties. The CEH is not a fixed surface in space but a state of the spacetime fluid defining the causal boundary.
Classical GR is the hydrodynamic (the low energy, long wavelength) regime of a more fundamental microscopic theory of spacetime.
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