Tail bounds for detection times in mobile hyperbolic graphs

We first introduce the model of Krioukov et al. [28] in its Poissonized version (see also [21] for the same description in the so called uniform model): for each $n\in\mathbb{N}^{+}$, consider a Poisson point process $\mathcal{P}$ on the hyperbolic disk of radius $R:=2\log(n/\nu)$ for some positive constant $\nu\in\mathbb{R}^{+}$ ($\log$ denotes here and throughout the paper the natural logarithm). The intensity function $\mu$ at polar coordinates $(r,\theta)$ for $0\leq r\leq R$ and $-\pi\leq\theta<\pi$ is equal to $nf(r,\theta)$, where $f(r,\theta)$ is given by

$\displaystyle f(r,\theta)$

$\displaystyle:=\begin{cases}\displaystyle\frac{\alpha\sinh(\alpha r)}{2\pi(\cosh(\alpha R)-1)},&\text{if $0\leq r\leq R$},\\[8.61108pt] 0,&\text{otherwise.}\end{cases}$

In other words, the angle and radius are chosen independently; the former uniformly at random in $(-\pi,\pi]$ and the latter with density proportional to $\sinh(\alpha r)$.

Next, identify the points of the Poisson process with vertices and define the following graph $G_{n}:=(V_{n},E_{n})$ where $V_{n}:=\mathcal{P}$. For $P,P^{\prime}\in V_{n}$, $P\neq P^{\prime}$, with polar coordinates $(r,\theta)$ and $(r^{\prime},\theta^{\prime})$ respectively, there is an edge in $E_{n}$ with endpoints $P$ and $P^{\prime}$ provided the hyperbolic distance $d_{H}$ between $P$ and $P^{\prime}$ is such that $d_{H}\leq R$, where $d_{H}$ is obtained by solving

$$\cosh d_{H}:=\cosh r\cosh r^{\prime}-\sinh r\sinh r^{\prime}\cos(\theta{-}\theta^{\prime}).$$

(1)

In particular, note that $\mathbb{E}(|V_{n}|)=n$.

Henceforth, we denote the point whose radius is $0$ by $O$ and refer to it as the origin. For a point $P$ and $r\geq 0$ we let $B_{P}(r)$ denote the ball centered at $P$ of radius $r$, that is, the set of all points at hyperbolic distance less than $r$ from $P$. Also, we henceforth denote the boundary of $B_{P}(r)$ by $\partial B_{P}(r)$.

To define a dynamic version of Krioukov et al.’s model we consider an initial configuration $\mathcal{P}$ of the Poissonized model in $B_{O}(R)$ and then associate to each $x_{0}:=(r_{0},\theta_{0})\in\mathcal{P}$ a particle that evolves independently of other particles following a trajectory given in radial coordinates by $x_{t}:=(r_{t},\theta_{t})$ at time $t$. At a microscopic level a natural choice for the movement of a particle is that of a random walk in $B_{O}(R)$ with $\partial B_{O}(R)$ acting as a reflecting boundary, and where at each step the particle can move either in the radial or angular direction. Assuming that the movement does not depend on the angle and that there is no angular drift, we conclude that at a macroscopic level particles should move according to a generator of the form

$$\Delta_{h}=\frac{1}{2}\frac{\partial^{2}}{\partial r^{2}}+\frac{\alpha}{2}\frac{1}{\tanh(\alpha r)}\frac{\partial}{\partial r}+\frac{1}{2}\sigma^{2}_{\theta}(r)\frac{\partial^{2}}{\partial\theta^{2}},$$

where the drift term in the radial component is chosen so that $f(r)drd\theta$ remains the stationary distribution of the process (this can be checked using the Fokker-Planck equation). The function $\sigma^{2}_{\theta}(\cdot)$ is unrestricted and relates to the displacement given by the angular movement at a given position $(r,\theta)$. In this sense a natural choice is to take $\sigma^{2}_{\theta}(r)$ proportional to $\sinh^{-2}(r)$ which follows by taking the displacement proportional to the hyperbolic perimeter at that radius. An alternative, however, is to take $\sigma^{2}_{\theta}(r)$ proportional to $\sinh^{-2}(\alpha r)$ corresponding to the (re-scaled) Brownian motion in hyperbolic space. In order to capture both settings, we introduce an additional parameter $\beta$ and work throughout this paper with the following generalized generator:

$$\Delta_{h}:=\frac{1}{2}\frac{\partial^{2}}{\partial r^{2}}+\frac{\alpha}{2}\frac{1}{\tanh(\alpha r)}\frac{\partial}{\partial r}+\frac{1}{2\sinh^{2}(\beta r)}\frac{\partial^{2}}{\partial\theta^{2}}$$

(2)

where $\beta>0$ is a new parameter related to the velocity of the angular movement which can alternatively be understood as a deformation of the underlying space. We shall see that thus enhancing our model yields a broader range of behavior and exhibits phase transition phenomena (for detection times this is explained in detail in our next section where we summarize our main results).

Fix now an initial configuration of particles $\mathcal{P}$ located at points in $B_{O}(R)$. We denote by $\mathbb{P}_{x_{0}}$ the law of a particle initially placed at a given point $x_{0}:=(r_{0},\theta_{0})\in B_{O}(R)$. We have one more fixed target $Q$, located at the boundary of $B_{O}(R)$ (that is, $r_{Q}=R$), and at angular coordinate $\theta_{Q}:=0$. For any $s>0$, let $\mathcal{P}_{s}\subseteq\mathcal{P}$ be the set of points that have detected the target $Q$ by time $s$: that is, for each $P\in\mathcal{P}_{s}$ there exists a time instant $0\leq t\leq s$, so that $x_{t}\in B_{Q}(R)$. Note that if $t=0$, then $P$ might be in the interior of $B_{Q}(R)$, whereas if $t>0$, the first instant at which $P$ detects $Q$ is when $P$ is at the boundary of $B_{Q}(R)$, that is $x_{t}\in\partial B_{Q}(R)$. This instant $t$ is also called the hitting time of $B_{Q}(R)$ by particle $P$. The detection time of $Q$, denoted by $T_{det}$, is then defined as the minimum hitting time of $B_{Q}(R)$ among all initially placed particles. We are particularly interested in the tail probability $\mathbb{P}_{x_{0}}(T_{det}\geq\mathfrak{z})$ for different values of $\mathfrak{z}$ (note that $\mathfrak{z}$ is a function on $n$). In words, we are interested in the tail probability that no particle initially located at position $x_{0}$ detects the target $Q$ by time $\mathfrak{z}$.

Observe that any given particle $P$ evolving according to the generator $\Delta_{h}$ specified in (2) will eventually detect the target at some point, so that $\mathbb{P}(T_{det}\geq\mathfrak{z})\to 0$ when $\mathfrak{z}\to\infty$ as soon as there is at least one particle. Following what was done in [37] for a similar model on Euclidean space, our main result determines the speed at which $\mathbb{P}(T_{det}>\mathfrak{z})$ tends to zero as a function of $\mathfrak{z}$. We consider the same setting of [37], that is $\mathfrak{z}/\mathbb{E}(T_{det})\to\infty$ but we have to deal with several additional, both qualitatively and quantitatively different, new issues that arise due to the dependency of $\mathfrak{z}$ on $n$.

In this section, we present the main results we obtain regarding tail probabilities for detection time, both for the mobile hyperbolic graph model we introduced in the previous section and for two restricted instances: one where the radial coordinate of particles does not change over time and another where the angular coordinate does not change. We also discuss the relation between the main results and delve into the insights they provide concerning the dominant mechanism (either angular movement, radial movement or a combination of both) that explain the different asymptotic regimes, depending on the relation between $\alpha$ and $\beta$. We point out that we do not present in this section a complete list of other significant results, in particular the ones that provide a detailed idea of the initial location of particles that typically detect the target. These last results will be discussed at the start of Sections 3, 4, and 5 where, in fact, we prove slightly stronger results than the ones stated in this section. Neither do we delve here into those results concerning one-dimensional processes with non-constant drift and a reflecting barrier that might be useful in other settings and thus of independent interest (these results are found in Section 4).

We begin with the statement describing the (precise) behavior of the detection time tail probability depending on how the model parameters $\alpha$ and $\beta$ relate to each other:¹¹1We use the standard Bachmann–Landau asymptotic (in $n$) notation of $O(\cdot)$, $o(\cdot)$, $\Omega(\cdot)$, $\omega(\cdot)$, $\Theta(\cdot)$, with all terms inside asymptotic expressions being positive.

Observe that by Theorem 1, for $\mathfrak{z}=\Theta((e^{\beta R}/n)^{2})$ in the case $\beta\leq\frac{1}{2}$, and for $\mathfrak{z}=\Theta(1)$ in the case $\beta>\frac{1}{2}$ we recover a tail exponent of order $\Theta(1)$, showing that the expected detection time occurs at values of $\mathfrak{z}$ of said order. It follows that at $\beta=\frac{1}{2}$ there is a phase transition in the qualitative behavior of the model; for $\beta<\frac{1}{2}$, since $(e^{\beta R}/n)^{2}=o(1)$, the detection becomes asymptotically “immediate” whereas for $\beta>\frac{1}{2}$ the target can remain undetected for an amount of time of order $\Theta(1)$. To explain this change in behavior notice that even though for any $\beta>0$ the value of $\sigma^{2}_{\theta}(r):=\sinh^{-2}(\beta r)$ is minuscule near the boundary of $B_{O}(R)$ (where particles spend most of the time due to the radial drift), decreasing $\beta$ does increase $\sigma_{\theta}(\cdot)$ dramatically, allowing for a number of particles tending to infinity, to detect the target immediately. Since for small values of $\mathfrak{z}$ all but a few particles remain “radially still” near the boundary, we deduce that there must be some purely angular movement responsible for the detection of the target, to which we can associate the tail exponent $ne^{-\beta R}\sqrt{\mathfrak{z}}$ seen in (i) of Theorem 1. The same tail exponent appears in Case (iic) of Theorem 1 for large $\mathfrak{z}$ when $\beta$ is again sufficiently small (smaller than $\frac{\alpha}{2}$), and again the purely angular movement is responsible for the detection of the target. To make the understanding of the observed distinct behaviors even more explicit we study two simplified models where particles are restricted to evolve solely through their angular or radial movement, respectively.

Observe that the tail exponent appearing in the case $\alpha>\beta$ of the previous theorem is the same as the one appearing in Case (iic) (where $\beta\leq\frac{1}{2}<\alpha$) and Case (i) (where $\beta<\frac{\alpha}{2}<\alpha$) of Theorem 1: this is no coincidence. Our proof shows that a purely angular movement is responsible for detection in those cases. Note also that the other exponents contemplated in Theorem 1 are not present in Theorem 3.

We consider next the result of our analysis of the second simplified model where particles move only radially:

Once more observe that the tail exponent in the $\beta>\frac{1}{2}$ case of Theorem 1 is the same as the one in Theorem 4, and once more this is not a coincidence: our proof shows that when $\beta>\frac{1}{2}$ the detection of the target is the result of the radial motion of particles alone. In contrast, Cases (iia) and (iib) in Theorem 1, when $\frac{1}{2}<\alpha\leq 2\beta\leq 1$ yield new tail exponents not observed neither in Theorem 3 nor in Theorem 4, and which is larger than those given in said results. Since a larger tail exponent means a larger probability of an early detection, in this case, neither the angular nor the radial component of the movement dominates the other, but they rather work together to detect the target quicker: in Case (iia), the proof reveals that the radial movement is responsible for pulling particles sufficiently far away from the boundary to a radial value, where $\sigma_{\theta}(r):=\sinh^{-2}(\beta r)$ becomes sufficiently large (although still small) so that with a constant probability at least one particle has a chance of detecting at this radial value through its angular motion. Finally, for Case (iib) in Theorem 1 we see a tail exponent of the form $ne^{-\beta R}\sqrt{\mathfrak{z}}$ (as expected when taking $\alpha\nearrow 2\beta$ in Case (iia) or $\alpha\searrow 2\beta$ in Case (iic)) but accompanied by a logarithmic correction. This correction becomes clear when inspecting the proof: it is a result of the balance between the effect of the radial drift and the contribution of the angular movement of particles at distinct radii whose effect is that the contributions to the total angular movement coming from the time spent by a particle within a narrow band centered at a given radius are all about the same, adding up and giving the additional logarithmic factor.

Our main theorem, that is Theorem 1, provides insight on how unlikely it is for the target $Q$ to remain undetected until time $\mathfrak{z}$, and as discussed before, the proof strategy as well as our remarks provide additional information about the dominant detection mechanisms (that is, either by angular movement only, by radial movement only, or by a combination of the two). Moreover, through our techniques we can obtain more precise information showing in each case where particles must be initially located in order to detect $Q$ before time $\mathfrak{z}$ with probability bounded away from zero. Specifically, given a parameter $\kappa>0$ (independent of $n$) we can construct a parametric family of sets $\mathcal{D}^{(\mathfrak{z})}(\kappa)$ depending on $\mathfrak{z}:=\mathfrak{z}(n)$ and the parameters of the model, such that for every $x_{0}\in\mathcal{D}^{(\mathfrak{z})}(\kappa)$ and sufficiently large $n$, the probability $\mathbb{P}_{x_{0}}(T_{det}\geq\mathfrak{z})$ is at least a constant that depends on the parameter $\kappa$. Even further, we will show that $\mu(\mathcal{D}^{(\mathfrak{z})}(\kappa))$ is of the same order as the tail exponents, which implies that asymptotically the best chance for $Q$ to remain undetected by time $\mathfrak{z}$ is to find these sets initially empty of points. To be precise, we show the following meta-result:

The added value of this last theorem is that it reveals that the set $\mathcal{D}^{(\mathfrak{z})}(\kappa)$ can roughly be interpreted as the regions where particles typically detecting the target are initially located. Moreover, each of the Theorems 1, 3, and 4 follow from the instances of Theorem 5 regarding angular, radial and unrestricted movement, respectively, thus revealing the unified approach that we take throughout the paper as explained in detail in Subsection 1.3.

Establishing the uniform lower bounds for the different cases of Theorem 5 requires, especially in Case (iib) of Theorem 1, a delicate coupling with a discretized auxiliary process. For the integral upper bounds a careful analysis has to be performed: we thereby need to calculate hitting times of Brownian motions in the hyperbolic plane with a reflecting barrier where we make use of results on tails of independent sums of Pareto random variables (that differ greatly according to the exponent).

Finally, let us point out that from a technical point of view, the additional difficulties we encounter compared to Euclidean space are substantial: in [37] the authors address the Euclidean analogue of the problem studied in this paper by relating the detection probability to the expected volume of the $d$-dimensional Wiener sausage (that is, the neighborhood of the trace of Brownian motion up to a certain time moment – see also the related work section below). To the best of our knowledge, however, the volume of the Wiener sausage of Brownian motion with a reflecting boundary in hyperbolic space is not known. Even further, our proof techniques give significantly more information: with our proof we are able to characterize the subregions of the hyperbolic space such that particles initially located therein are likely to detect the target up to a certain time moment, whereas particles located elsewhere are not - with the information of the pure volume of the Wiener sausage we would not be able to do so. The radial drift towards the boundary of $B_{O}(R)$ entails many additional technical difficulties that do not arise in Euclidean space (see also the related work section below for work being done there). Moreover, our setup contains a reflecting barrier at the boundary, thereby adding further difficulty to the analysis.

 

Recall that in our mobile hyperbolic graph model we consider an initial configuration $\mathcal{P}$ of the Poissonized model in $B_{O}(R)$ and then associate to each $x_{0}:=(r_{0},\theta_{0})\in\mathcal{P}$ a particle that evolves independently of other particles according to the generator $\Delta_{h}$ with a reflecting barrier at $\partial B_{0}(R)$. Denote by $\mathcal{P}_{\mathfrak{z}}$ the Poisson point process obtained from $\mathcal{P}$ by retaining only particles having detected the target by time $\mathfrak{z}$. Since points move independently of each other, each particle initially placed at $x_{0}\in B_{O}(R)$ belongs to $\mathcal{P}_{\mathfrak{z}}$ independently with probability $\mathbb{P}_{x_{0}}(T_{det}\leq\mathfrak{z})$ and hence $\mathcal{P}_{\mathfrak{z}}$ is a thinned Poisson point process on $B_{O}(R)$ with intensity measure

$$d\mu_{\mathfrak{z}}(x)\;=\;\mathbb{P}_{x_{0}}(T_{det}\leq\mathfrak{z})d\mu(x).$$

Noticing that the event $\{T_{det}\geq\mathfrak{z}\}$ is equivalent to $\{\mathcal{P}_{\mathfrak{z}}=\emptyset\}$, we have

$$\mathbb{P}(T_{det}\geq\mathfrak{z})=\exp\Big{(}-\int_{B_{O}(R)}\mathbb{P}_{x_{0}}(T_{det}\leq\mathfrak{z})d\mu(x_{0})\Big{)}$$

(3)

and hence in order to obtain lower and upper bounds on $\mathbb{P}(T_{det}\geq\mathfrak{z})$ we will compute the dependence of this integral as a function of $\mathfrak{z}$. In fact, we can say a bit more on the way we go about determining the value of the said integral. Specifically, we partition the range of integration $B_{O}(R)$ into $\mathcal{D}^{(\mathfrak{z})}(\kappa)$ and $\overline{\mathcal{D}}^{(\mathfrak{z})}(\kappa)$ and observe that the three parts of Theorem 5 together immediately imply that

$$\int_{B_{O}(R)}\mathbb{P}_{x_{0}}(T_{det}\leq\mathfrak{z})d\mu(x_{0})=\Theta(\mu(\mathcal{D}^{(\mathfrak{z})}(\kappa)))$$

thus reducing, via (3), the computation of tail bounds for detection times to fixing the parameter $\kappa$ equal to a constant and determining the expected number of particles initially in $\mathcal{D}^{(\mathfrak{z})}(\kappa)$, that is, computing $\mu(\mathcal{D}^{(\mathfrak{z})}(\kappa))$. So, the structure of the proof of our main results is the following: we determine a suitable candidate set $\mathcal{D}^{(\mathfrak{z})}(\kappa)$, compute $\mu(\mathcal{D}^{(\mathfrak{z})}(\kappa))$, and establish an adequate version of Theorem 5 for the specific type of movement of particles considered (angular, radial or a combination of both).

After the introduction of random hyperbolic graphs by Krioukov et al. [28], the model was then first analyzed in a mathematically rigorous way by Gugelmann et al. [21]: they proved that for the case $\alpha>\frac{1}{2}$ the distribution of the degrees follows a power law, they showed that the clustering coefficient is bounded away from $0$, and they also obtained the average degree and maximum degree in such networks. The restriction $\alpha>\frac{1}{2}$ guarantees that the resulting graph has bounded average degree (depending on $\alpha$ and $\nu$ only): if $\alpha<\frac{1}{2}$, then the degree sequence is so heavy tailed that this is impossible (the graph is with high probability connected in this case, as shown in [9]). The power-law degree distribution of random hyperbolic graphs is equal to $2\alpha+1$ (see [21, Theorem 2.2]) which, for $\alpha\in(\frac{1}{2},1]$, belongs to the interval $(2,3]$, and this is the interval where the best fit power-law exponent of social networks typically falls into (see [5, p. 69]). If $\alpha>1$, then as the number of vertices grows, the largest component of a random hyperbolic graph has sublinear size (more precisely, its order is $n^{1/(2\alpha)+o(1)}$, see [8, Theorem 1.4] and [14]). On the other hand, it is known that for $\frac{1}{2}<\alpha<1$, with high probability a hyperbolic random graph of expected order $n$ has a connected component whose order is also linear in $n$ [8, Theorem 1.4] and the second largest component has size $\Theta(\log^{\frac{1}{1-\alpha}}n)$ [24], which justifies referring to the linear size component as the giant component. More precise results including a law of large numbers for the largest component in these networks were established in [17]. Further results on the static version of this model include results on the diameter [23, 19, 34], on the spectral gap [25], on typical distances [1], on the clustering coefficient [12, 18], on bootstrap percolation [26] and on the contact process [29].

No dynamic model of random hyperbolic graphs has been proposed so far. In the Boolean model (that is similar to random geometric graphs but now the underlying metric space is $\mathbb{R}^{d}$ and, almost surely, there is an infinite number of vertices) the same question of the tail probability of detection time, as well as the questions of coverage time (the time to detect a certain region of $\mathbb{R}^{d}$), percolation time (the time a certain particle needs to join particle of the infinite component) and broadcast time (the time it takes to broadcast a message to all particles) were studied by Peres et al. [37]: the authors therein show that for a Poisson process of intensity $\lambda$, and a target performing independent Brownian motion, as $t\to\infty$,

$$\mathbb{P}{(T_{det}\geq t)}=\exp\Big{(}-2\pi\lambda\frac{t}{\log t}(1+o(1))\Big{)},$$

where a target is detected if a particle is at Euclidean distance at most $r$, for some arbitrary but fixed $r$ (in the three other mentioned contexts the interpretations are similar: we say that an interval is covered if each point of the interval has been at distance at most $r$ from some particle at some time instant before $t$; we say that a particle joins a component if it is at distance at most $r$ from another particle of the component, and we say that a message can be sent between two particles if they are at distance $r$). Note that the probability holds only as $t\to\infty$, and in this case the main order term of the tail probability does not depend on $r$, as shown in [37]. Stauffer [39] generalized detection of a mobile target moving according to any continuous function and showed that for a Poisson process of sufficiently high intensity $\lambda$ over $\mathbb{R}^{d}$ (with the same detection setup) the target will eventually be detected almost surely, whereas for small enough $\lambda$, with positive probability the target can avoid detection forever. The somewhat inverse question of isolation of the target, that is, the time it takes for a (possibly) moving target (again for a Poisson process of intensity $\lambda$ in $\mathbb{R}^{d}$) until no other vertex is at distance $r$ anymore and the target becomes isolated, was then studied by Peres et al. [36]: it was shown therein that the best strategy for the target to stay isolated as long as possible in fact is to stay put (again with the same setup for being isolated).

Also for the Boolean model, the question of detection time was already addressed by Liu et al. [30] when each particle moves continuously in a fixed randomly chosen direction: they showed that the time it takes for the set of particles to detect a target is exponentially distributed with expectation depending on the intensity $\lambda$ (where detection again means entering the sensing area of the target). For the case of a stationary target as discussed here, as observed in Kesidis et al. [22] and in Konstantopoulos [27] the detection time can be deduced from classical results on continuum percolation: namely, in this case it follows from Stoyan et al. [40] that

$$\mathbb{P}{(T_{det}\geq t)}=e^{-\lambda\mathbb{E}(vol(W_{r}(t)))},$$

where $vol(W_{r}(t))$ is the volume of the Wiener sausage of radius $r$ up to time $t$ (equivalent to being able to detect at distance at most $r$), and which in the case of Euclidean space is known quite precisely [38, 6].

A study of dynamic random geometric graphs prior to the paper of Peres et al. was undertaken by Díaz et al. [13]: the authors therein consider a random geometric graph whose radius is close to the threshold of connectivity and analyze in this setup the lengths of time intervals of connectivity and disconnectivity. Even earlier, the model of Brownian motion in infinite random geometric graphs (in the ”dynamic Boolean model”) was studied by van den Berg et al. [7] who showed that for a Poisson intensity above the critical intensity for the existence of an infinite component, almost surely an infinite component exists at all times (here the radii of particles are not fixed to $r$, but rather i.i.d. random variables following a certain distribution). More generally, the question of detecting an immobile target (or escaping from a set of immobile traps) when performing random walks on a square lattice is well studied (here, in contrast to the previous papers, detecting means to be exactly at the same position in the lattice); escaping from a set of mobile traps was recently analyzed as well, for both questions we refer to Athreya et al. [4] and the references therein.

Section 2 gathers facts and observations that will be used throughout the paper. Section 3 then analyzes the simplified model where the particles’ movement is restricted to angular movement only, at the same time illustrating in the simplest possible setting our general proof strategy. Section 4 then considers the model where the particles’ movement is restricted to radial movement only. In Section 5 we finally address the mixed case, that is, the combined case of both radial and angular movement of particles. Specifically, in Section 5.1, we look at quick detection mechanisms and, in Section 5.2, we show that the detection mechanisms found are asymptotically optimal in all cases, that is, no other strategy can detect the target asymptotically faster. In Section 6 we briefly comment on future work.

Throughout all of this paper $\alpha$ is a fixed constant such that $\frac{1}{2}<\alpha\leq 1$ (the reason to consider only this range is that only therein hyperbolic random graphs are deemed to be reasonable models of real-world networks – see the discussion in Section 1.4). Furthermore, in this article both $\beta$ and $\nu$ are strictly positive constants. In order to avoid tedious repetitions and lengthy statements, we will not reiterate these facts. Also, wherever appropriate, we will hide $\alpha$, $\beta$, and $\nu$ within asymptotic notation. Moreover, throughout the paper, we let $\mathfrak{z}:=\mathfrak{z}(n)$ be a non-negative function depending on $n$.

In order to avoid using ceiling and floors, we will always assume $R:=2\ln(n/\nu)$ is an integer. Since all our claims hold asymptotically, doing so has no impact on the stated results. All asymptotic expressions are with respect to $n$, and wherever it makes sense, inequalities should be understood as valid asymptotically.