Is ‘being above the median’ a noise sensitive property?

In the most well-studied setting, first-passage percolation is the study of the random metric space that arise by assigning non-negative independent random weights, from some common distribution $F$, to the edges of the ${\mathbb{Z}}^{2}$ nearest-neighbour lattice. For simplicity, we shall in this paper stick to the planar setting, and we shall for most of the paper assume that $F$ is supported on $\{a,b\}$ for some $0<a<b<\infty$. The edge weights induce a metric $T$ on ${\mathbb{Z}}^{2}$ as follows: For $u,v\in{\mathbb{Z}}^{2}$, set

The infimum in $T(u,v)$ is known to be attained for some finite path, although this path does not have to be unique. We let $\pi(u,v)$ denote this path, and apply some deterministic rule for selecting one in case it is not unique.

For $n\geq 1$, set $T_{n}:=T(0,n{\bf e}_{1})$ and $\pi_{n}:=\pi(0,n{\bf e}_{1})$ for brevity, where ${\bf e}_{1}:=(1,0)$ denote the first coordinate vector. A standard consequence of the Subadditive Ergodic Theorem is the existence of a constant $\mu$, known as the time constant, such that almost surely

Also $\pi_{n}$ is known to be of linear order, although it is not known to have a well-defined asymptotic speed. In approaching the problem of the current paper, one soon requests a finer description of the order of fluctuations, both for $T_{n}$ around its mean, and $\pi_{n}$ away from the coordinate axis. Predictions from the physics literature [33], which have been established for related models of last-passage percolation [8, 32], suggest that fluctuations of $T_{n}$ are order $n^{1/3}$ and transversal fluctuations of $\pi_{n}$ are order $n^{2/3}$.

The approach proposed by Itai Benjamini, in September 2011, to explore questions of noise sensitivity in the context of first-passage percolation (and for other real-valued observables) can be described as follows: The distance $T_{n}$ is a random variable, whose distribution may be unknown. This distribution has a median $m_{n}$, and for large $n$, this median can be expected to split the distribution of $T_{n}$ roughly in half, i.e. that

Under the assumption that the weight distribution $F$ is supported on $\{a,b\}$, for some $0<a<b<\infty$, the event that $T_{n}$ exceeds its median can be encoded as a Boolean function. If (3) holds, then this function is non-degenerate. It is thus possible, as proposed by Benjamini, to investigate whether $T_{n}$ exceeding its median is a noise sensitive property, within the framework of Boolean functions.

In this paper we shall present the first progress on Benjamini’s problem. We have not been able to answer the question as it has been formulated above, for reasons that we shall elaborate upon below. As stated the question thus remains open. Indeed, although (3) trivially holds for continuous weight distributions, it seems to remain unknown whether, uniformly in $n$,

for some median of $T_{n}$, when $F$ is discrete. See, however, [15, 21] for results in this direction.

In order to circumvent the difficulties faced above, we shall make two simplifications to Benjamini’s problem. First, we replace point-to-point passage times by horizontal crossing times of squares, and hence increase the symmetry in the problem. Second, we restrict the transversal fluctuations allowed by paths crossing the squares. Given $k\geq 1$, let $\mathcal{P}_{k}(n)$ denote the set of nearest-neighbour paths contained the ‘square’ $[0,n]\times[0,n-1]$ that connect the left side to the right, and whose vertical displacement is at most $k$, and set

Our main result is the following theorem, which makes the first progress on Benjamini’s problem. Our result is formulated for an arbitrary quantile of the crossing variable, and not just its median.

We remark that the analogous result holds for the square replaced by an $n\times n$-torus, and $\tau(n,k_{n})$ replaced by the minimal weight among all circuits crossing the torus horizontally, and whose transversal fluctuations are bounded by $k_{n}$. Moreover, while we here focus on the planar setting, an analogous statement holds also in dimensions $d\geq 3$, with a stronger restriction on the growth of $k_{n}$. In both cases, adapting the proof of Theorem 1.1 is straightforward.

Related to the study of noise sensitivity is the notion of ‘chaos’, that stems from the physics literature on spin-glasses [13, 26]. In the context of first-passage percolation, chaos refers to the sensitivity of the distance-minimising path $\pi_{n}$ as opposed to the distance $T_{n}$. The first rigorous evidence of chaos was obtained by Chatterjee in two preprints [16, 17], later combined into a book [18]. That the first-passage metric is chaotic was established only recently, in work of Ahlberg, Deijfen and Sfragara [3]. To state this result, let $\pi_{n}^{\varepsilon}$ denote the distance-minimising path between the origin and $n{\bf e}_{1}$ with respect to the perturbed weights $\omega^{\varepsilon}$. If, for instance, $F$ is continuous and has finite moment of order $2+\log$, then

Significantly more precise results, determining the rate at which $\varepsilon=\varepsilon(n)$ is allowed to decay with $n$, have been obtained by Ganguly and Hammond [27] for certain integrable models of last-passage percolation; see also [4] for related results.

Our result differ from the above in that it addresses the sensitivity of the metric $T$ as opposed to the structure minimising $T$, and (to our knowledge) this result is the first of its kind for a (supercritical) spatial growth model. Note, however, related work of Damron, Hanson, Harper and Lam [20] that establish the existence of exceptional times in a dynamical version of critical first-passage percolation.

A key result from the original paper of Benjamini, Kalai and Schramm [10] gives a criterion for a sequence of Boolean functions to be noise sensitive in terms of the notion of influences. The influence of bits is central in computer science, and has its origin in social choice theory. The influence of a bit $i\in\{1,2,\ldots,n\}$ of a function $f:\{0,1\}^{n}\to\{0,1\}$ is defined as the probability that the bit is desisive for the outcome of the function, i.e.

where $\sigma_{i}:\{0,1\}^{n}\to\{0,1\}^{n}$ is the operator that flips the entry at position $i$. The criterion, which has come to be known as the BKS Theorem, states that if

then the sequence $(f_{n})_{n\geq 1}$ is noise sensitive.

Apart from the computation of influences, as the BKS Theorem invites to, there are other methods by which noise sensitivity may be established. The main development has occurred with applications to Bernoulli percolation in mind: A method involving the revealment of randomised algorithms was developed by Schramm and Steif [38]; The Fourier spectrum of critical percolation was analysed by Garban, Pete and Schramm [28]; A probabilistic approach was taken by Tassion and Vanneuville [39], inspired by Kesten’s scaling relations. Neither of these routes seem easy to follow in our context. Moreover, for monotone functions (which we are concerned with here) the criterion in (6) is both necessary and sufficient for a sequence to be noise sensitive. So, either directly or indirectly, verifying (6) is inevitable. This will, hence, be the route we take.

Let us start with a general observation. For functions $f:\{0,1\}^{n}\to{\mathbb{R}}$ that are Lipschitz, i.e. have bounded differences $|f(\omega)-f(\sigma_{i}\omega)|\leq c$ for some constant $c>0$ and all $i$, if changing the value of a bit $i$ takes $f$ from below to above its median $m$ (or vice versa), then $f$ must have been within distance $c$ from the median. In particular, we have the distributional bound

Standard variance bounds for functions that are Lipschitz (a.k.a. having bounded differences) with constant $c$ give $\textup{Var}(f)\leq c^{2}n$; see [12, Corollary 3.2]. Hence the above distributional bound gives a bound of order $1/\sqrt{n}$ at best. This would amount to an upper bound on the sum of influences squared being a non-vanishing constant. Hence, it cannot in general be sufficient to bound the influences simply considering the distribution of $f$.

Note that, regardless if we consider $T_{n}$ or $\tau(n,k)$, changing the value of an edge may affect the observable by at most $\pm(b-a)$, meaning that they are both Lipschitz. Hence, a simple distributional bound as in (7) will not suffice. Using an observation from [11], we may link influential edges to edges on the geodesic. Recall that $\pi_{n}$ is the path (a path in case of multiple) attaining the infimum in $T_{n}$. Then,

The predictions from KPZ universality suggest that $|T_{n}-m_{n}|\leq b-a$ should occur with probability order $n^{-1/3}$, and that a typical edge being on the geodesic has probability order $n^{-2/3}$. For most edges within distance $n^{2/3}$ or the coordinate axis the influence is thus order $1/n$, and for edges further away it is negligible. This amounts to a bound on the sum of influences squared that vanishes with $n$.

The above heuristic is merely conjectural, and we are nowhere close to establish statements like this in first-passage percolation. It is generally not even known whether

for some median $m_{n}$ of $T_{n}$. However, a result by Pemantle and Peres [37] implies such a statement for exponentially distributed edge weights.

In a first attempt to simplify Benjamini’s problem it is tempting to replace the point-to-point passage times with the left-right crossing times of rectangles. Let $\mathcal{P}(n,m)$ denote the set of nearest-neighbour paths contained in the rectangle $R(n,m):=[0,n]\times[0,m-1]$ that connect the left side to the right. We define the crossing time of the rectangle $R(n,m)$ as

In particular, we let $t_{n}:=t(n,n)$ denote the crossing time of the ‘square’ $R(n,n)$. The increasing level of symmetry attained in this way is manifested in that all horizontal and all vertical bonds of the square an effect of $t_{n}$ that is roughly equal.³³3Admittedly, the square may have to be replaced by a torus, and the crossing time of the square by the circumference of the torus, for this to be fully rigorous. From the linear upper bound on the length of a geodesic due to Kesten [34], a bound analogous to (8) would give

and hence

This shows that even if we spread out the contribution coming from ‘being on the geodesic’, only considering the contribution from the geodesic, and not the distributional properties of the crossing time, will not suffice in order to deduce noise sensitivity.

Again, it remains unknown whether the probability in the right-hand side of (10) vanishes as $n\to\infty$. In fact, also the weaker question whether the variance of $t_{n}$ diverges as $n\to\infty$ remains unknown; the best lower bound gives a constant. We refer the reader to the recent work of Damron, Houdré and Özdemir [22] for a further discussion in this direction.

We shall circumvent the above mentioned difficulties in calculating the influences by imposing a restriction on the transversal fluctuations of the paths admissible for crossing the square $R(n,n)$.

The restriction on transversal fluctuations does not result in a lower asymptotic velocity by which the square is crossed, as long as the allowed fluctuations diverge with $n$; see [1, 14]. That is, for any diverging sequence $(k_{n})_{n\geq 1}$ we have, almost surely,

However, it is expected that the restriction does have an effect on a lower order. For $k$ fixed, on the other hand, one may show that there exists $\mu_{k}>\mu$ such that almost surely

To see how the transversal restriction will help in calculating the influences, let us consider the case when $k=1$. Note that $\tau(n,1)$ is the sum of $n$ independent binomial random variables with parameters $n$ and $1/2$. Using moderate deviation estimates of the binomial distribution we are able to compute the asymptotic behaviour of the quantiles of their minimum as well as the influences. Note how the case $k=1$ is reminiscent of the classical Tribes function, introduced by Ben-Or and Linial [9], but with polynomial-sized tribes as opposed to logarithmic. We treat the case $k=1$ in detail in Section 2, as special case of a larger family of polynomial Tribes functions.

For $k\geq 2$ we may express $\tau(n,k)$ as the minimum of $n-k+1$ identically distributed, but dependent, variables as follows. Let $R_{i}(n,k)$ denote the rectangle $[0,n]\times[i,i+k-1]$, and $t_{i}(n,k)$ the horizontal crossing time of $R_{i}(n,k)$. Since every path in $\mathcal{P}_{k}(n)$ may fluctuate vertically at most $k$, it has to be contained in $R_{i}(n,k)$ for some $i=0,1,\ldots,n-k$. It follows that

For fixed $k$, the distribution of these variables is asymptotically Gaussian, as proved (in parallel) by Ahlberg [1] and Chatterjee and Dey [14]. In fact, the latter paper shows that the asymptotic normality continues to hold for $k=k(n)$ growing slower than $n^{1/3}$. The asymptotic normality will not be sufficient in itself, as we will need to peak into the tail of the distribution, in that $\tau(n,k)$ is a minimum of a large number of variables. For that reason, we shall need to combine the approach from [14] with a Cramér-type moderate deviations theorem for triangular arrays (Theorem 3.1), in order to obtain a moderate deviations theorem for first-passage percolation across thin rectangles (Theorem 4.1). With the moderate deviations estimates at hand, we will be able to approximate the asymptotic behaviour of quantiles and influences for the restricted crossing time $\tau(n,k)$, and prove Theorem 1.1.

We remark that the asymptotic normality is in itself not central to our approach. The relevant part is that it allows us to bound the influence of an edge by a rare enough event, whose probability we may compute. As a by-product of our proof we obtain the following estimate on the fluctuations on $\tau(n,k)$.

While we here focus on weight distributions supported on two points, we remark that our proof of the above theorem goes through without change for bounded weight distributions. Apart from a result by Pemantle and Peres [37] for exponentially distributed edge weights, it remains unknown whether for every $c>0$ we have

It would be interesting to establish (11) for a large class of weight distributions.

The analogous problem for geodesics is the well-known ‘midpoint problem’, which was posed by Benjamini, Kalai and Schramm in [11]. Interestingly, this problem has been solved for continuous weight with finite mean by Ahlberg and Hoffman [6]. Their result shows that for every edge $e$ we have

Earlier work of Damron and Hanson [19] gave a conditional proof under plausible, but unverified, assumptions on the asymptotic shape. In more recent work, Dembin, Elboim and Peled [23] derive polynomial rates on the decay in (12) for a more restrictive class of weight distributions. However, as mentioned above, without progress on the problem in (11), these results are insufficient for making further progress on Benjamini’s problem.

The rest of this paper is organised as follows. In Section 2 we showcase out approach by considering a polynomial version of the classical Tribes function. In Section 3 we prove a Cramér-type moderate deviations theorem for triangular arrays. In Section 4 we apply the moderate deviations theorem to prove a moderate deviations theorem for first-passage percolation across thin rectangles, which will allow us to analyse the asymptotic behaviour of the quantiles of our main observable. In Section 5 we derive a preliminary version of our main theorem, in which we consider the minimum crossing time across disjoint rectangles. Finally, in Section 6, we prove our main results, and in Section 7 we elaborate upon some open problems.