Smoothness and monotonicity of the excursion set density of planar Gaussian fields

Nazarov and Sodin have shown that the number of connected components of the nodal set of a planar Gaussian field in a ball of radius $R$, normalised by area, converges to a constant. This has been generalised to excursion/level sets at arbitrary levels, implying the existence of functionals $c_{ES}(\ell)$ and $c_{LS}(\ell)$ that encode the density of excursion/level set components at the level $\ell$. We prove that these functionals are continuously differentiable for a wide class of fields. This follows from a more general result which derives differentiability of the functionals from the decay of the probability of `four-arm events' for the field conditioned to have a saddle point at the origin. For some fields, including the important special cases of the Random Plane Wave and the Bargmann-Fock field, we also derive stochastic monotonicity of the conditioned field, which allows us to deduce regions on which $c_{ES}(\ell)$ and $c_{LS}(\ell)$ are monotone.


Introduction
Let f : R 2 → R be a continuous stationary Gaussian field with zero mean and covariance function K : R 2 → R defined by K(x) = E(f (x)f (0)). We are interested in the geometric properties of the (upper-)excursion sets and level sets of this field, defined respectively as x ∈ R 2 : f (x) ≥ and x ∈ R 2 : f (x) = for ∈ R. Recent work has established that, in many circumstances, the geometry of these sets exhibits similar behaviour to discrete percolation models [4]. In particular, for a wide class of f it has been shown that the connectivity of the excursion sets exhibits a sharp phase transition at = 0 [21,25]. A basic geometric property of these sets is the number of connected components in a large domain. Unlike certain other geometric functionals (e.g. the volume or Euler characteristic of the excursion sets), this quantity is inherently difficult to study because it is non-local: the number of components in a domain cannot be counted by partitioning the domain and simply counting the number of components in each sub-domain. Nazarov and Sodin [22] used an ergodic argument to study the asymptotics of this quantity. Specifically, if f is an ergodic Gaussian field satisfying some regularity assumptions, B(R) is the ball of radius R > 0 centred at the origin, and N LS (R, 0) is the number of components of the nodal set {x ∈ R 2 : f (x) = 0} contained in B(R), then there is a constant c LS (0) ≥ 0 such that (1.1) N LS (R, 0)/(πR 2 ) → c LS (0) as R → ∞, where convergence occurs in L 1 and almost surely. Although this result was stated only for the nodal set, the arguments in [22] go through verbatim for excursion/level sets at arbitrary levels ; the respective limiting constants, denoted by c ES ( ) and c LS ( ), can be interpreted as the density of excursion/level set components per unit area.
In this paper we consider properties of c ES ( ) and c LS ( ) viewed as functions of the level. It was shown in [6] that c ES and c LS are absolutely continuous. Our main results (Theorems 2.10, 2.11 and Corollary 2.18) show that, for a wide class of fields, the continuous differentiability of c ES and c LS at is equivalent to the statement that, if the field is conditioned to have a saddle point at the origin at level , then almost surely the 'arms' of the saddle (i.e. the four level lines that emanate from the saddle point) do not connect the origin to infinity. Since we can prove that the latter property holds for many fields, we deduce the continuous differentiability of the density functionals.
Our result can be compared to what is known, and conjectured, about the analogous density functional for discrete percolation models. Consider Bernoulli bond percolation on the integer lattice, defined by declaring the edges of the integer lattice Z d to be open independently with probability p and closed otherwise (see [15] for background on this model). Let K n denote the number of open clusters that are contained in [−n, n] d . Then it is known ( [15,Chapter 4]) that as n → ∞, almost surely and in L 1 . This is a direct analogue of (1.1), and is also proven using an ergodic argument. The smoothness of κ is of interest because it is related to the percolation phase transition. Specifically, it is conjectured in the physics literature that κ is analytic on [0, 1]\{p c } and twice but not three times differentiable at p c , where p c ∈ (0, 1) is the critical probability for the model; this reflects the values of certain 'critical exponents' which are believed to be universal for percolation models (see [15,Chapter 9]). What has been shown rigorously is that, for all d ≥ 2, κ is analytic on [0, p c ) and smooth on (p c , 1], and in the case d = 2 it is further known that κ is analytic on (p c , 1] and twice differentiable at p c (see [15,Chapter 4]). Somewhat weaker results have been derived for other percolation models, including the Poisson-Boolean model and 'spread-out' percolation models [9].
Since the connectivity of the excursion sets of a wide class of planar Gaussian fields is conjectured, and in some cases known, to undergo a phase transition at = 0 that is analogous to the phase transition at p c for Bernoulli percolation (see [4,7,21,24,25]), it is natural to conjecture that, for such fields, c ES and c LS are also analytic on R\{0} and twice but not three times differentiable at 0. Our proof of the continuous differentiability of c ES and c LS can be seen as a first step in this direction.
Despite the connections to classical percolation theory, the method we use to prove differentiability of the density functionals is quite different. In Bernoulli percolation, the starting point is the equality where |C| is the number of vertices in the open cluster at the origin. By enumerating clusters, this can be expressed as a power series in p, and the smoothness of κ can be deduced from bounds on the coefficients in terms of connection probabilities for the cluster at the origin.
This approach does not readily generalise to the setting of Gaussian fields: whilst it can be shown that where Vol(C) is the volume of the component of {x ∈ R 2 : f (x) ≥ } containing the origin, it is not known whether the density of (Vol(C), f (0)) is jointly continuous ([8] studies a kind of 'ergodic' density for Vol(C) at the zero level). Instead, our proof of differentiability uses an integral representation for c ES and c LS that was developed in [6] (see Theorem 2.6), although we still rely on the decay of certain 'connection probabilities' for the field f conditioned to have a saddle point at the origin. These connections are the equivalent of 'four-arm events' in percolation, which play an important role in this theory (e.g., in the analysis of noise sensitivity [14]). Our study of the integral representation for c ES and c LS also allows us to derive certain montonicity properties of these functionals (see ; these results are of independent interest, and are a key input to proving lower bounds on the variance of the number of excursion/level sets of Gaussian fields (see Remark 2.24).

Main results
Throughout the paper we consider a planar Gaussian field satisfying the following assumption: Assumption 2.1. The Gaussian field f : R 2 → R is continuous, centred, and stationary. The spectral measure µ, defined by is a (Hermitian) probability measure, is not supported on two lines through the origin, and satisfies R 2 |t| 4+η dµ(t) < ∞ for some η > 0. Since µ is a probability measure, Var(f (x)) = 1 for all x ∈ R 2 . The measure µ not being supported on two lines through the origin is equivalent to the Gaussian vector 1 ∇ 2 f (x) being non-degenerate (Lemma A.1). The moment condition on µ ensures that f ∈ C 2+η (R 2 ) almost surely for any η ∈ (0, η /4) (Kolmogorov's theorem for two-dimensional fields [3, Proposition 1.16]) and we fix such an η for our analysis.
We have in mind two important examples of Gaussian fields satisfying Assumption 2.1: (1) The Random Plane Wave (RPW), with covariance K(x) = J 0 (|x|) where J 0 is the 0-th Bessel function, and spectral measure equal to the normalised Lebesgue measure on the unit circle; and (2) The Bargmann-Fock (BF) field, with covariance K(x) = exp(−|x| 2 /2), and Gaussian spectral measure. The RPW is a universal model for high energy eigenfunctions of the Laplacian, see [10] for background. The BF field can be viewed as a continuous analogue of Bernoulli percolation, since it has rapid correlation decay and satisfies the FKG inequality, see [4] for details and further motivation.
We now formally define the density functionals c ES and c LS . Let N ES,R ( ) and N LS,R ( ) denote respectively the number of components of {x ∈ R 2 : f (x) > } and {x ∈ R 2 : f (x) = } contained in B(R) (i.e. the components which intersect B(R) but not R 2 \B(R)). Then the following asymptotic laws are known to hold: 6,19,22]). Let f be a Gaussian field satisfying Assumption 2.1. For each The constants implied by the O(·) notation are independent of . If f is also ergodic, then almost surely and in L 1 .
Remark 2.3. The notation in [6] and some other papers is slightly different: c LS in the present paper is denoted by c N S in some previous papers.
In [6] a representation of c ES and c LS was given in terms of the densities of certain types of critical points. To state this we introduce upper/lower connected saddle points.
Definition 2.4. Let x 0 be a saddle point of a C 2 function g : R 2 → R such that there are no other critical points at the same level as x 0 (that is, if x 1 is another critical point of g, then g(x 1 ) = g(x 0 )). We say that x 0 is upper connected if it is in the closure of only one component It was shown in [12,11] that the expected number of local maxima, local minima or saddle points of a Gaussian field with height in a certain range can be expressed as the integral of an explicit continuous density function over the height range. In [6] this result was extended to upper and lower connected saddle points without explicitly computing the corresponding density functions: Proposition 1.7]). Let f be a Gaussian field satisfying Assumption 2.1. Then there exist non-negative functions p m + , p m − , p s + , p s − , p s ∈ L 1 (R) such that the following holds. Let Ω ⊂ R 2 be compact and ∂Ω have finite Hausdorff-1 measure. Let ∈ R and let N m + , N m − , N s + , N s − and N s denote the number of local maxima, local minima, upper connected saddles, lower connected saddles and saddles of f in Ω with level above respectively. Then for h = m + , m − , s + , s − , s. Furthermore, these functions can be chosen to satisfy the relations p m + (x) = p m − (−x), p s + (x) = p s − (−x) and p s − + p s + = p s , and such that p m + , p m − and p s are continuous.
We can now state the main result of [6], characterising c ES and c LS in terms of the densities in Proposition 2.5: Theorem 1.8]). Let f be a Gaussian field satisfying Assumption 2.1, and let p m + , p m − , p s + , p s − denote the densities specified in Proposition 2.5. Then and hence c ES and c LS are absolutely continuous.
One of the motivations for Theorem 2.6 was to provide a tool with which to study the excursion/level set densities: since p m + , p m − , and p s = p s + + p s − are explicitly known for a wide class of fields, by establishing simple properties of p s − we can deduce corresponding properties for c ES and c LS . We expand upon this method in this paper. More precisely, we consider the function p * s − ( ) := p s ( ) P f has a lower connected saddle point at the origin , wheref is the field f conditioned to have a saddle point at the origin at level (in the sense of Palm distributions; see Lemma 3.2 for a formal definition). Under mild conditions we show that p * s − defines a version of p s − (recall that the latter is defined only up to null sets). By studyingf we are able to deduce properties of p * s − , and hence of c ES and c LS . 2.1. Differentiability. Our first set of results concerns the differentiability of c ES and c LS . Let us begin by detailing the necessary assumptions on f .
is non-degenerate.
Moreover, there exists a neighbourhood V of the origin on which the spectral measure µ has density ρ with respect to the Lebesgue measure and inf V ρ > 0. Assumption 2.9. For 0 < r < R, let Arm (r, R) denote the 'one-arm event' that there exists a component of {f ≥ } which intersects both ∂B(r) and ∂B(R). Then there exist c 1 , c 2 > 0 such that for any 1 < r < R Assumption 2.7 is extremely mild; it is satisfied whenever the support of the spectral measure µ contains an open set or an open interval of a non-degenerate conic section (Lemma A.2). In particular, it holds for the RPW and the BF field. Assumptions 2.8 and 2.9 are somewhat more restrictive. Assumption 2.8 holds for any smooth field with sufficiently nice correlation decay, and in particular holds for the BF field, but it does not hold for the RPW (whose correlations decay only as |t| −1/2 ). It also implies Assumption 2.7 by the previous remark. Assumption 2.9 relates to the conjectured properties of the 'percolation universality class', and has been shown to hold for a wide class of fields that includes the BF field [25]. Moreover it is strongly believed to hold for the RPW. We state our results directly in terms of one-arm decay as it is likely that these bounds will be extended to more fields over time.
Our first main result is that c ES and c LS are continuously differentiable under the above assumptions: Theorem 2.10. Suppose f is a Gaussian field satisfying Assumptions 2.1 and 2.7-2.9. Then c ES and c LS are continuously differentiable on R. In other words, the functions p s − and p s + defined in Proposition 2.5 can be chosen to be continuous, and We emphasise that Theorem 2.10 applies to a wide class of fields, including the important case of the BF field, but does not apply to the RPW as stated (although we believe the conclusion to be true).
2.1.1. Four-arm saddle points. Theorem 2.10 follows from a more general result establishing that, under very mild conditions, the continuous differentiability of c ES and c LS is implied by the decay of certain connection probabilities involving 'four-arm saddles'.
Let D ⊂ R 2 be a simply connected domain with piecewise C 1 boundary and let x 0 ∈ D be a saddle point of g ∈ C 2 (R 2 ) such that g has no other critical points at the same level as x 0 . We say that x 0 is four-arm in D if it is in the closure of two components of {x ∈ D : g(x) > g(x 0 )} and two components of {x ∈ D : g(x) < g(x 0 )} (see Figure 1a); intuitively, a saddle point is four-arm in D if we cannot tell whether it is upper or lower connected by looking at the values of g in D. A saddle point x 0 is said to be infinite four-arm if it is in the closure of two components of {x ∈ R 2 : g(x) > g(x 0 )} and two components of {x ∈ R 2 : g(x) < g(x 0 )} (see Figure 1b). As mentioned in Section 1, four-arm saddle points are analogous to four-arm events for percolation models.    Then p * s − | (a,b) is continuous, and hence c ES and c LS are continuously differentiable on (a, b). Theorem 2.10 follows from Theorem 2.11 once we verify condition (2.5) under Assumptions 2.8 and 2.9. To do so we use Assumption 2.8 and a Cameron-Martin argument to treat the conditional fieldf away from the origin as a perturbation of the unconditioned field f . We then use Assumption 2.9 to bound the relevant connection probabilities for the unconditioned field.
As a corollary of Theorem 2.11 (actually of its proof), we deduce a bound on the number of saddle points of a Gaussian field that are four-arm inside a ball and whose level lies in a narrow range. This improves a bound that was previously established in [6], and is also a key ingredient in the recently derived lower bounds on the variance of the number of excursion/level set components (see Remark 2.24).
Corollary 2.12. Let f be a Gaussian field satisfying all the assumptions of Theorem 2.11. Then there exists a function δ R → 0 as R → ∞ and a constant c > 0 such that, for each Remark 2.13. In [6] it was shown that E (N 4-arm (R)) = O(R); Corollary 2.12 supersedes this bound whenever b R − a R = O(R −1 ). It is possible to improve the conclusion of Corollary 2.12 further by imposing stronger assumptions on the field. For example, suppose we assume the exponential decay of arm probabilities at non-zero levels: for all * > 0 and δ ∈ (0, 1), there exist c 1 , c 2 > 0 such that Then for any a > * (or b < − * ), it is possible to prove that there exists c > 0 such that In [21], it is shown that a wide class of fields satisfy (2.6) so this assumption is reasonable. We do not prove this result formally here because Proposition 2.12 is simpler to prove, holds for a wider class of fields, and suffices for its intended purpose (see Remark 2.24). In light of Theorem 2.11, the fact that (2.6) is expected to hold for a wide class of fields also suggests it should be much easier to prove differentiability of c ES away from zero, since the probability of four-arm saddles in B(R) should decay exponentially at non-zero levels.
2.1.2. The positivity of the level set density. In order for Theorem 2.2 to describe the leadingorder asymptotics of the number of excursion/level set components, it is crucial that the limiting constants are positive; if they are not, then it can be shown that f almost surely has no compact excursion/level sets. One nice consequence of the differentiability of c ES and c LS is that it gives a new, short proof of their positivity in the delicate case = 0: Proposition 2.14. Let f be a Gaussian field satisfying Assumption 2.1. Suppose either c ES or c LS is continuously differentiable at 0. Then c ES (0) > 0 and c LS (0) > 0.
The positivity of c ES (0) and c LS (0) were already known under Assumption 2.1 ( [16,22] gave a variety of sufficient conditions, whose union can be checked to exhaust Assumption 2.1). We restate this result because it uses a very different method of proof; in particular, it does not rely on the 'barrier method'.
The positivity of c ES ( ) and c LS ( ) for > 0 is simpler to establish, even without differentiability, see [6]. On the other hand, our arguments apparently do not extend to c ES ( ) for < 0 (although this case can still be treated via the 'barrier method'; see Lemma 2.23).

2.1.3.
Fields outside the 'percolation universality class'. Although in general we expect the properties of c ES and c LS to match those of the analogous density functional κ from percolation theory, this can fail for fields outside the 'percolation universality class'.
To demonstrate this, we consider the one non-trivial case in which c ES and c LS are explicitly known: fields with spectral measure supported on four or five points (see [6,Proposition 3]). In [6] it was shown that, in the 'five point case', c ES and c LS are smooth everywhere, whereas in the 'four point case', c ES and c LS are smooth everywhere except zero, at which point they are continuous but not differentiable (see Figure 2). Hence, in both cases, the smoothness of c ES and c LS differs from the conjectured properties of κ (and in different ways). However, these fields do not fall within the scope of the present paper (they do not satisfy Assumptions 2.1 and 2.7). Moreover, being periodic, their large-scale properties cannot be expected to match those of Bernoulli percolation. On the other hand, the non-differentiability of c ES at zero in the 'four point case' does reflect a different kind of phase transition: for ≤ 0, {f ≥ } almost surely has no bounded components (c ES ( ) = 0), whereas for > 0, the number of components is of order R 2 (c ES ( ) > 0); see Figure 3. Moreover, a Gaussian field in the 'five point case' can be represented as a field in the 'four point case' plus an independent Gaussian level shift. Hence the same phase transition occurs, although it does so at a random level and so the discontinuity is averaged out. Figure 3. Stylised excursion sets for fields with spectral measure supported on four points, at the zero level (left) and at a positive level (right).

2.2.
Monotonicity. We next consider monotonicity properties of c ES and c LS . We begin by analysing the ratio p * s − ( )/p s ( ) = P f has a lower connected saddle point at the origin , which we intuitively expect to be non-decreasing: if we condition on the origin being a saddle point at increasing heights, it seems more likely that it should be lower connected. This can be made rigorous under some additional assumptions, and allows us to deduce regions on which c ES ( ) and c LS ( ) are monotone.
Assumption 2. 15. The field f is isotropic (that is, its law is invariant under rotation), the Gaussian vector (f (0), ∇ 2 f (0)) is non-degenerate, and for all x ∈ R 2 , In Section 5.1.2 we explain how (2.7) can be translated into an explicit property of the conditional fieldf . We can also give an equivalent version of (2.7) that is easier to check in practice. Since we assume that f is isotropic, its covariance function may be expressed as K(x) = k(|x| 2 ). If this function is rescaled so that k (0) = −1, then it can be shown by Gaussian regression that (2.7) is equivalent to From this, it can be verified that specific fields satisfy this assumption, including the BF field and the field with covariance K(x) = (1 + (β/2)|x| 2 ) −β/2 for β ≥ 1. The RPW does not satisfy Assumption 2.15, however we are able to prove monotonicity of p * s − /p s in this case too: Theorem 2.16. Let f be the Random Plane Wave or a field satisfying Assumptions 2.1, 2.7 and 2.15. Then p * s − ( )/p s ( ) is non-decreasing in . Given the definition of p * s − , Theorem 2.16 is an immediate consequence off − being stochastically decreasing in . Our proof of this differs for the RPW and for fields satisfying Assumption 2.15 (in the former case it is somewhat simpler, because of the degeneracies in the RPW; see e.g. [28]).
The monotonicity of p * s − /p s has some immediate consequences for the smoothness of c ES and c LS : 17. Let f be the Random Plane Wave or a field satisfying Assumptions 2.1, 2.7 and 2.15. Then p * s − has at most a countable set of discontinuities, all of which are jump discontinuities. In particular, c ES and c LS are twice differentiable almost everywhere.
Another consequence of monotonicity is that it implies a converse of Theorem 2.11: 18. Let f be the Random Plane Wave or a field satisfying Assumptions 2.1, 2.7 and 2.15. Then for every a < b the following are equivalent: (1) For all ∈ (a, b) P f has an infinite four-arm saddle at 0 = 0; (2) There exists a version of p s − which is continuous on (a, b); Remark 2.19. Clearly, if any of (1)-(4) hold in Corollary 2.18, then by Theorem 2.11, the version of p s − | (a,b) which is continuous is equal to p * s − | (a,b) . Finally we use Theorem 2.16 to deduce intervals on which c ES and c LS are monotone. We shall state the strongest form of our results only in the case of the RPW and BF field. Let D + and D + respectively denote the lower and upper right Dini derivatives, that is, for g : R → R, Proposition 2.20. Let f be the Random Plane Wave. Then and We also present weaker results for general isotropic fields satisfying Assumption 2.15. Recall that the covariance function of an isotropic f may be expressed as [11] for details on this parameter).
As an intermediate result to Proposition 2.20 we require that, for the RPW, c ES ( ) > 0 for ≤ 0. Since this result is not stated elsewhere in the literature, we do so here. The proof uses the 'barrier method' and is near-identical to that in [22] in the case = 0.
Remark 2.24. Many of the results in this work will be built upon by [5] in order to prove lower bounds on the variance of the number of level/excursion set components in B(R) as R → ∞. Specifically, it will be shown that if f has sufficiently nice correlation decay (such as the BF field), and if c ES has a non-zero derivative at , then for some c > 0 and all R sufficiently large. Moreover, if f is the RPW and one of the Dini derivatives of c ES is non-zero for = 0, then for some c > 0 and all R sufficiently large. Analogous results hold in both cases for level sets and c LS . A key step in proving these results is to estimate the order of , which is made possible by Theorem 2.10 and Corollary 2.12. Since the lower bounds also require that c ES has a non-zero derivative/Dini derivative at , Propositions 2.20-2.22 are crucial for ensuring that they are widely applicable.

2.3.
Outline of the remainder of the paper. In Section 3 we give a formal definition off , the field f conditioned to have a saddle point at the origin at level , and derive explicit distributions forf in special cases. In Section 4 we study topological properties off , and use this to deduce the results outlined in Section 2.1. In Section 5 we consider stochastic monotonicity properties off , and complete the proofs of the results in Section 2.2. Appendix A contains miscellaneous results on the non-degeneracy of Gaussian fields.

The field conditioned to have a saddle at the origin
In this section we considerf , the field f conditioned to have a saddle point at the origin at level . Using the theory of Palm distributions we give an explicit representation forf , and in the isotropic case we derive simple expressions for its distribution.
We begin with a general statement expressingf as (a limit of) a Palm distribution relative to a point process defined by the saddle points of f . Let us first recall the relevant theory of Palm distributions (see [18,Chapter 11] for background). Let g : R 2 → R be a planar random field and S a non-degenerate, simple point process on R 2 , and suppose that (g, S) are jointly stationary. Fix a bounded Borel set B ⊂ R 2 . Then the Palm distribution of g relative to S is defined as the random fieldg satisfying, for any Borel set A of finite-dimensional projections, . This definition is independent of the reference set B, and so we may writeg = (g | {0} ∈ S). It is important to distinguishf [ , + ] from the conditioned field which is defined via the distributional limit The latter is sometimes known as 'vertical window conditioning', and is the standard way of conditioning on part of a random vector (for a Gaussian vector, this conditioning is given explicitly by Gaussian regression, see [3, Proposition 1.2]). By constrast, the former can be thought of as 'horizontal window conditioning', and corresponds to sampling a 'typical' saddle point (i.e. via the counting measure). The difference between these forms of conditioning is elegantly explained in [17]. Using basic properties of Gaussian fields, we can derive explicit representations forf [ , + ] andf : where g is a centred Gaussian field with covariance function γ, and z [ , + ] , Z [ , + ] is an independent random vector with density 3 The functions α, β and γ in Lemma 3.2 can be computed explicitly via Gaussian regression (see [3, and let Σ 0 and Σ be the respective covariance matrices of these vectors. Then In the case that (f (0), ∇ 2 f (0)) is degenerate (which includes the RPW), the representations off [ , + ] andf in Lemma 3.2 must be modified to accommodate this degeneracy; in particular, ∇ 2 f (0) should be considered as a vector consisting of two of its coordinates, chosen so that they are non-degenerate with f (0), and α, β and γ defined accordingly. For simplicity we will not state this representation formally, since for the RPW we state a more precise description below (in Proposition 3.4).
Lemmas 3.1 and 3.2 are essentially derived in [2, Chapter 6]; we repeat this here for completeness, and so that we can extend the the arguments slightly. ; which form we are using will always be clear from context. We also use this convention for Z and X (introduced below).

By inspecting their joint distribution, it is clear that
. We now fix a sequence i ↓ 0, and create a coupling off [ , + i ] for each i such that each field consists of the same realisation of g and the sequences {z [ , + i ] } i∈N and {Z [ , + i ] } i∈N converge almost surely. Since K ∈ C 2+η (R 2 ), the same is true of α, β and γ and hence g ∈ C 2+η (R 2 ) almost surely for the choice of η ∈ (0, η /4) made at the beginning of Section 2 (Kolmogorov's theorem [3,Proposition 1.16]). It is therefore clear that the coupled fieldsf [ , + i ] converge almost surely in the C 2+η topology uniformly on compact sets as i → ∞ tof . This completes the proof of the lemmas.
We now present simpler descriptions forf in the case of isotropic fields. In this case it is quite natural to express the Hessian component Z in terms of its eigenvalues λ 1 < λ 2 and the direction of the first eigenvector θ. Recall the parameter χ = −k (0)/ k (0) ∈ (0, √ 2], where K(t) = k(|t| 2 ). Again we must distinguish the case in which (f (0), ∇ 2 f (0)) is degenerate, which corresponds to χ = √ 2 and implies that f is (a rescaled version of) the RPW.
3. Let f be an isotropic field satisfying Assumption 2.1 such that χ < √ 2. Thenf where g, α and β are as in Lemma 3.2, θ is an independent random variable uniform on [0, 2π), and (λ 1 , λ 2 ) is an independent random vector with density Proof. Recall the random vector Z from Lemma 3.2, which we view as a 2 × 2 symmetric matrix. Let λ 1 < λ 2 be the eigenvalues of Z , and let θ be the angle of the eigenvector associated to λ 1 . If h denotes the bijection which maps Z to Λ := (λ 1 , λ 2 , θ), then for any Borel set A Since f is isotropic and (f (0), ∇ 2 f (0)) is non-degenerate, [11] derives the density of the ordered eigenvalues of (∇ 2 f (0)|f (0) = ) and their direction as that given above.
where g is a centred Gaussian field with covariance function γ (defined as in Lemma 3.2), α, β are defined as and Z = (Z 11 , Z 12 ) t is an independent random vector with density In particular, the one-point distribution off (t) is where (θ, λ) is an independent random vector with density follows from an argument similar to that used to prove Lemma 3.2. The functions α and β can be explicitly calculated using Gaussian regression (see [3, Proposition 1.2] for example).
Next we note that (Z 11 , Z 12 ) is supported on the region for which Therefore this matrix almost surely has a unique, positive eigenvalue λ and direction θ. By explicitly diagonalising this matrix, we obtain a formula for λ and θ: Z 11 + /2 = (λ + /2) cos(2θ) By the standard change of variable formula (and explicitly evaluating ψ in terms of the covariance of the RPW) we can calculate the joint density of (λ, θ) to be equal to the expression in (3.3). To evaluate the one-point distribution, we simply set t 2 = 0.

Differentiability of excursion/level set functionals
In this section we prove the results stated in Section 2.1. We begin by studying the space C 2+η Reg of functions h ∈ C 2+η R 2 which have a non-degenerate critical point at the origin and no other critical points at level h(0). We will also use the space C 2+η Reg (R) which is the set of all h ∈ C 2+η Reg such that h(0) is not a critical level of h| ∂B(R) . We endow these spaces with the C 2+η loc topology.
By showing thatf ∈ C 2+η Reg (R) almost surely, we prove thatf having an upper (or lower) connected saddle point in a compact region is a continuity event, from this we deduce Theorem 2.11 (with the other results following as consequences).  Proof. Let h ∈ C 2+η Reg (R) have a saddle point at the origin which is R-lower connected; we will find a neighbourhood around h which contains only functions with such saddle points at the origin. First we choose r ∈ (0, 1) sufficiently small that h has a four-arm saddle in B(r). Since the origin is a non-degenerate saddle point for h, ∇ 2 h(0) has eigenvalues λ 1 < 0 < λ 2 and corresponding eigenvectors v 1 , v 2 . We now choose a neighbourhood N 1 ⊂ C 2+η Reg (R) of h (in the topology of uniform C 2+η convergence) such that for all g ∈ N 1 , This ensures that each function in N 1 also has a saddle point at the origin.
Next we choose N 2 ⊂ C 2+η Reg (R) such that for each g ∈ N 2 , g 2+η ≤ 2 h 2+η . We consider the four line segments joining 0 to ∂B(r) parallel to v 1 and v 2 and we reduce r relative to h 2+η so that for each g ∈ N 1 ∩ N 2 , the directional derivative of g on this line segment (parallel to the line segment) has constant sign. This ensures that for each such g the saddle point at the origin is four-arm in B(r).
There exist two connected subsets A 1 , A 2 of ∂B(r) such that h < h(0) − 3 on A 1 ∪ A 2 for some > 0 and A 1 and A 2 are in different components of B(r) ∩ {h < h(0)} (see Figure 4). We next choose a neighbourhood N 3 ⊂ C 2+η Reg (R) of h such that A 1 and A 2 have the same properties for any function g ∈ N 3 with 3 replaced by 2 and |g(0) − h(0)| < .
By definition of a saddle being R-lower connected, there is a curve γ in B(R) joining A 1 to A 2 in {h < h(0)} and h is bounded above by h(0) − 3δ on γ for some δ > 0. Since γ is compact we can find a neighbourhood N 4 such that g < h(0) − 2δ on γ for all g ∈ N 4 and |h(0) − g(0)| < δ. Combining these observation, we see that N : Reg (R)) and any g ∈ N has a saddle point at the origin which is lower connected in B(R) and so the set of functions with such saddle points is open, as required.
The set C 2+η Reg can be partitioned into sets of functions which have either a local maxima, a local minima, a saddle point which is four-arm in B(R) or a saddle point which is Rupper/lower connected at the origin. Arguments which are very similar to those above show We next confirm thatf ∈ C 2+η Reg (R) almost surely: 3. If f is a Gaussian field satisfying Assumptions 2.1 and 2.7, then for any ∈ R and R > 0,f ∈ C 2+η Reg (R) almost surely.
Proof. To simplify the presentation we shall assume that (f (0), ∇ 2 f (0)) is non-degenerate, since the degenerate case is similar. Recall the representation off in Lemma 3.2. By the definitions of α, β and γ,f is almost surely in C 2+η (R 2 ), and has a critical point at the origin at level . By evaluating the second order derivatives of α, β and γ, it follows that ∇ 2f (0) = Z . Since the density of Z is identically zero on the region where its determinant is zero, det ∇ 2f (0) = 0 almost surely, and so the critical point at the origin is non-degenerate. Next we show thatf almost surely has no other critical points at level . Let T n = B(n)\B( 1 n ) and consider (∇f ,f − ) : T n → R 3 . Bulinskaya's lemma ([1, Lemma 11.2.10]) states thatf almost surely has no critical points at level in T n provided the univariate densities of (∇f (t),f (t)) are bounded in a neighbourhood of (0, ) uniformly over t ∈ T n . Since g and Z are independent, the density of (∇f (t),f (t)) is given by the convolution Therefore, to show that p ∇f (t),f (t) is bounded, it is sufficient to show that the density of (∇g(t), g(t)) is bounded uniformly in t. Since these densities are Gaussian, this is equivalent to showing that the determinant of the covariance matrix of (∇g(t), g(t)) is bounded away from 0 on T n . However this is the determinant of Cov (f (t), ∇f (t)) f (0), ∇f (0), ∇ 2 f (0) which is non-degenerate for each t ∈ T n by Assumption 2.7. Since this determinant is continuous in t, it is bounded away from 0 on the compact set T n . Taking the countable union of T n for n ∈ N then shows thatf almost surely has no critical points at level in R 2 \{0}.
We are now ready to prove Theorem 2.11. Let N (R) s − is replaced with N s − or N s , we make a corresponding definition for lower connected saddle points or saddle points respectively. Recall that s − (R) is the subset of functions in C 2+η Reg (R) with an R-lower connected saddle point at the origin. We also define s − and s + to be the subsets of C 2+η Reg with lower and upper connected saddle points at the origin respectively.
Proof of Theorem 2.11. Let f be a field satisfying Assumptions 2.1 and 2.7. The first step is to show that p * s − is lower semi-continuous by expressing it as the pointwise supremum of a sequence of continuous functions. Let ∈ R and > 0 and we now fix R > 0. By the definition off [ , + ] (recall (3.1)) By Lemma 3.1,f [ , + ] converges in distribution tof (in the C 2+η loc topology) as → 0, and since having a saddle point at the origin which is R-lower connected is a continuity event for f (Lemmas 4.2 and 4.3), the portmanteau lemma implies that as → 0. By inspecting the form of p Z it is clear thatf d − →f 0 as → 0 in the C 2+η loc topology. So by applying the portmanteau lemma again, we see that P(f ∈ s − (R)) is continuous in . Hence the function s − ( ). We now allow R to vary; since s − (R) is non-decreasing in R for each , we deduce that p (R) s − is also non-decreasing in R for each . Noting that these densities are bounded above by p s ( ), we apply dominated convergence to (4.1) to verify that for each ∈ R. Hence p * s − is indeed a pointwise supremum of continuous functions, and so is lower semi-continuous.
We next prove that p * s − = p s − almost everywhere. Let A ⊂ R be any Borel set. Then since where in the last line we have used the definition of p s − , the fundamental theorem of calculus applied to E(N , and then dominated convergence once again. Taking n to infinity shows that p * s − = p s − almost everywhere on A, and since A was arbitrary we conclude that these densities are equal almost everywhere. To finish the proof we show that the condition (2.5) thatf does not have an infinite fourarm saddle, implies the continuity of p * s − . Observe that, by repeating the arguments above, we may define the lower semi-continuous function which is a version of p s + . By Lemmas 4.1 and 4.3, the saddle point off at the origin must be either upper connected, lower connected or an infinite four-arm saddle. Therefore (Note that p s ( ) > 0 by Lemma A.3.) Now suppose that (2.5) holds, i.e., for all ∈ (a, b), f almost surely does not have an infinite four-arm saddle point at the origin. By (4.2) we see that p * s + ( ) = p s ( ) − p * s − ( ) for all ∈ (a, b). Since p * s + is lower semi-continuous (and p s is continuous), we deduce that p * s − is upper semi-continuous on (a, b). Hence we have shown that p * s − is both upper and lower semi-continuous on (a, b), which completes the result. As mentioned previously, Theorem 2.10 follows from Theorem 2.11 once we verify condition (2.5). This is done in the next lemma: Proof. By Assumption 2.8 and Lemma A.2 we know that (f (0), ∇ 2 f (0)) is non-degenerate. Recall the representation forf in Lemma 3.2 and recall also the explicit expressions for α, β and the covariance of g derived after this lemma. Since this covariance is expressed as the difference of two positive definite functions, if we let f 1 be a centred Gaussian field with covariance then we can decompose g = f − f 1 , where f 1 and f are independent. Since K 1 can be expressed as a linear combination of ∂ k 1 K(s)∂ k 2 K(t), for |k 1 |, |k 2 | ≤ 2, by Assumption 2.8 there exists c 1 , ν > 0 such that, for all r > 1, Moreover, since α, β can be expressed as a linear combination of ∂ k K(t), for |k| ≤ 2, by Assumption 2.8 there exists c 2 , ν > 0 such that Now, we apply a Cameron-Martin argument to the unconditional field f . In particular, by [21,Corollary 3.7] (valid by the assumption on the spectral density in Assumption 2.8, and since Arm (r, R) is an increasing event with respect to the field) there exists c 3 , r 0 > 0 such that for all r > r 0 the following holds: if F is a random field coupled with f such that We will apply this bound to F =f . Note that, by the union bound, We will show that, for a suitable choice of r = r R → ∞ and = R → 0, the three terms in (4.6) all decay to zero as R → ∞. By (4.4), and since Z ,i is almost surely finite, the first two terms in (4.6) converge to zero as long as r 1+ν → ∞. If we assume this convergence is sufficiently fast (to be specified below) then it is a standard estimate for the norm of a Gaussian field that the third term of (4.6) also converges to zero. This argument is essentially the same as [21,Lemma 3.12], but our setting is slightly different so we give a complete proof.
We turn to the third term. Let B x (1) denote the ball of radius 1 centred at x. Let A(r, R) denote the annulus of inner radius r and outer radius R. Covering A(r, R) with O(R 2 ) unit balls, and by the union bound, By the Borell-TIS inequality ([1, Theorem 2.1.1]), for all u > 0, where By Kolmogorov's theorem [22, A.9], there is a c 4 > 0 such that Taking u = /4 and assuming that /4 > c 5 r −1−ν we have To finish, we take r = R log(R) and = 1 R log(R) and observe that for this choice the right hand side of the estimate above converges to 0 as R → ∞. Combining all of these estimates together we have that the right hand side of (4.6) tends to zero as R → ∞.
Substituting into (4.5), and noting that r/R → 0 and R → 0 as R → ∞, proves that P(f ∈ Arm (r, R)) can be made arbitrarily small. Since this event is increasing in r, this completes the proof of the lemma.
Proof of Theorem 2.10. This is immediate from Theorem 2.11 and Lemma 4.4.
To end the section we prove the remaining results stated in Section 2.1, namely Corollary 2.12 and Proposition 2.14.
Proof of Corollary 2.12. By [6, Lemmas 1 and 4], E(N 4-arm (R)) = O(R) as R → ∞, so it suffices to prove the other bound here. Recall that A(R − r, R) denotes the annulus of inner radius R − r and outer radius R. We first note that for any 1 < r < R where, by a slight abuse of notation, for some c 1 , c 2 > 0 independent of R. By stationarity of f (4.8) s + are the continuous functions defined as in the proof of Theorem 2.11. In this proof it is shown that as r → ∞, p (r) s + converges pointwise monotonically to p * s − + p * s + = p s which is continuous. Therefore by Dini's theorem this convergence is uniform on [a, b] and so for any > 0 taking r sufficiently large relative to ensures that the right hand side of (4.8) is bounded above by R 2 (b R − a R ). If we choose r depending on R such that r → ∞ but r/R → 0 as R → ∞, then combining (4.7) and (4.8) proves the corollary.
Proof of Proposition 2.14. By Theorem 2.6 and the identities in Proposition 2.5, Let us also consider the function which can be shown via explicit calculation (see [6,Corollary 2]) to be equal to the C 1 function If c LS (0) = 0, then by the non-negativity of c ES it follows from (4.9) that c ES (0) = 0. Similarly, if c LS is continuously differentiable at 0, then by (4.9) and continuous differentiability of h, c ES is also continuously differentiable at 0.
It remains to show that if c ES is continuously differentiable at 0 then c ES (0) = 0. Suppose for the sake of contradiction that c ES (0) = 0. Then by the non-negativity of c ES , we have c ES (0) = 0. Hence, by (4.10), h ( ) = 0. Since h has critical points only at = ±1, we have derived the necessary contradiction.

Monotonicity results
In this section we prove the monotonicity results stated in Section 2.2. The main intermediate step is to show that the finite-dimensional projections off − are stochastically decreasing in , which we do in the next section. 5.1. Stochastic monotonicity. Our analysis differs depending on whether the field is the RPW or a general isotropic field satisfying Assumption 2.15, the RPW case being somewhat simpler. 5.1.1. Stochastic monotonicity for the RPW. Let f be the RPW. The first step is to show, via explicit calculation, thatf − is stochastically decreasing in at every point.
By Proposition 3.4,f has the distribution for the random vector (θ, λ) defined in that proposition. To simplify notation, we define The key fact leading to stochastic monotonicity is that, by Lemma 5.1 below, a = a(t, θ) ≥ 0 for all t and θ. This is equivalent to the statement that for all t ∈ R 2 For general isotropic fields, we show in Lemma 5.6 that Assumption 2.15 implies α(t) ≤ 1 (recall that α has a slightly different definition in the general case, see Lemma 3.2).
Remark 5.2. We prove this lemma by somewhat explicit computations that use sharp bounds on Bessel functions. We believe that there might be a more conceptual proof of this statement.
We shall actually show the slightly stronger statement thatf − is pointwise stochastically decreasing conditional on all values of (g, θ): Proof. Given the representation in (5.1), we have It remains to show that each of the expressions on the right-hand side of (5.2) are nondecreasing in for all values of g, θ and c. Recall that a = a(t, θ) ≥ 0 by Lemma 5.1. Hence 1 a +c−g(t)≥0 is clearly non-decreasing in . Moreover, after integrating (3.3), we see that for a differentiable function h : R → R where the last inequality follows from the fact that p λ (h) is zero unless h > 0 ∨ (− ). Now let h( ) = (c − g(t) + a )/b, and first suppose b > 0. Then h ( ) = a/b > 0, h/(2h + ) ≥ 0 on the region h > 0 ∨ (− ) and p λ (h) ≥ 0 (as a probability density) so (5.3) shows that the left hand side of (5.2) is non-decreasing whenever b > 0. Finally we suppose b < 0 and note that So once again, (5.3) shows the left hand side of (5.2) is non-decreasing whenever b < 0, completing the proof of the lemma.
We now extend this result to finite-dimensional projections off − . Recall that a random vector X = (X 1 , . . . , X n ) is said to stochastically dominate a random vector Y = (Y 1 , . . . , Y n ), written X Y , if E(g(X)) ≥ E(g(Y )) for any increasing g : R n → R. Clearly, if X Y then X i Y i for each i = 1, . . . , n. The converse is not true in general, but a useful sufficient condition can be formulated using the notion of copulas.
Theorem 5.4 (Theorem 1 of [27]). Let X = (X 1 , . . . , X n ) and Y = (Y 1 , . . . , Y n ) be random vectors with induced marginal probability measures P 1 , . . . , P n and Q 1 , . . . , Q n respectively. If Range(Q i ), and X i Y i for each i, then X Y . Using this theorem, we extend Lemma 5.3 to show the stochastic monotonicity of the finite-dimensional projectionsf − , conditional on any g, θ.
Lemma 5.5. For 1 < 2 and t 1 , . . . , t n ∈ R 2 , Proof. By Theorem 5.4 it is sufficient to show that the random vectors in (5.5) have the same copula (these copulas are uniquely defined on the same domain, and the stochastic domination of the marginals follows from the proof of Lemma 5.3). Fix ∈ R and t 1 , . . . , t n ∈ R 2 , and consider the copula By the definition of a and b, we can express for deterministic functions a, b. Since g(t i ), a(t i , θ) and b(t i , θ) are constants under the conditioning on (g, θ), and since copulas are invariant under strictly increasing transformations, where the last equality holds since g is independent of λ and θ. Notice that the random vector (λ · sign(b(t 1 , θ)), . . . , λ · sign(b(t 1 , θ)) θ) consists of λ multiplied by a constant vector (either of 1, −1 or 0). Hence by considering the alternative characterisation of a copula in (5.4), it is clear that Cop Z does not depend on the distribution of λ , and so is independent of .
for α, b 1 , b 2 as stated in the proposition (recall that b 1 and b 2 are defined in terms of β). The role of Assumption 2.15 is to ensure the following inequalities hold for α, b 1 , b 2 : Lemma 5.6. For all t ∈ R 2 and θ ∈ R, b 1 (t, θ) ≥ 0 , b 2 (t, θ) ≥ 0 and α(t) ≤ 1.
Proof. From the definition of b 1 (t, θ) and β, it is immediate that b 1 (t, 0) is the quantity given in Assumption 2.15 to be non-negative for all values of t. Since f is isotropic, b 1 is nonnegative for all values of θ. (By the identity cos(θ) = sin(θ + π/2), this also means that b 2 is non-negative.) We note that from the definition of α and β, α(t) = K(t) + 2k (0)(β 11 (t) + β 22 (t)).
Lemma 5.7. For any t 1 , . . . , t n ∈ R 2 and u 1 , . . . , u n ∈ R, Proof. Since the u i are arbitrary, we may assume g(t i ) = 0 for all i. We define the region It is enough to prove that the probability of the latter event is non-decreasing in because (λ 1 , λ 2 ) is independent of (g, θ).
Given the density of (λ 1 , λ 2 ), Since b 1 , b 2 ≥ 0 and α ≤ 1 by Lemma 5.6, A is non-decreasing in , and so for this derivative to be non-negative it is sufficient that By direct evaluation y), and since µ < 0, (5.6) is equivalent to . To complete the proof of the lemma, we show that this inequality holds for any possible region A . Since the shape of A might be quite complicated (see Figure 5 for a typical example), we divide the analysis into three cases and in each case show that conditioning on (λ 1 , λ 2 ) being contained in some simple region can only increase the expectation of λ 1 + λ 2 relative to conditioning on (λ 1 , λ 2 ) ∈ A .
x y A Figure 5. A typical example of the region A .
(Case 1). Suppose b 1 (t i , θ)/b 2 (t i , θ) = 1 for all i, so that the boundary of A , denoted ∂A , is a line of the form x + y = c for some c ∈ R. (Note that A is a subset of the entire plane not just the upper-left quadrant which is the support of (λ 1 , λ 2 ).) Then trivially and so (5.7) is verified in this case.
(Case 2). Now suppose that b 1 (t i , θ)/b 2 (t i , θ) ≥ 1 for all i and that this inequality is strict for some i (allowing for the degenerate case that b 2 (t i , θ) = 0). Let c * = E (λ 1 + λ 2 |(λ 1 , λ 2 ) ∈ A ) and then we note that the line x + y = c * must intersect ∂A at precisely one point (since the distribution of (λ 1 , λ 2 ) is continuous) which we denote by (d 1 , d 2 ). We now consider the region {x ≤ d 1 }, conditioning on (λ 1 , λ 2 ) lying in this region weakly increases the probability that λ 1 + λ 2 = c for c ≥ c * and weakly decreases this probability for c < c * (see Figure 6). Therefore x y x + y = c * (b) Figure 6. When A takes the form shown in 6a (Case 2), we condition on the region shown in 6b, which weakly increases the mean of λ 1 + λ 2 .
If b 1 (t i , θ)/b 2 (t i , θ) ≤ 1 for all i and this inequality is strict for some i, then an entirely analogous argument shows that for some d 2 . (Case 3). Suppose that for some i and j, b 1 (t i , θ)/b 2 (t i , θ) < 1 < b 1 (t j , θ)/b 2 (t j , θ). Defining c * as before we note that the line x+y = c * must intersect A (by definition of c * ) and so must intersect ∂A at two points (since the distribution of (λ 1 , λ 2 ) has no atoms). We denote these points by (d 1 , d 2 ) and (e 1 , e 2 ) and without loss of generality take d 1 < e 1 . We now consider the region {x < e 1 } ∩ {y < d 2 }. Reasoning as before, conditioning on (λ 1 , λ 2 ) lying in this region weakly increases the probability that λ 1 + λ 2 = c for c ≥ c * and weakly decreases this probability for c < c * (see Figure 7). So in this case . From (5.8), (5.9) and (5.10) we see that in order to complete the proof of the lemma, we need only verify (5.7) (or, equivalently, (5.6)) when A is of the form  in c. Applying this to y) dydx is non-increasing in c, and taking c → ∞ proves (5.7) for A = {λ 1 ≤ c}. It remains to verify that h 1 (x)/h 2 (x) is non-increasing in x, or equivalently, that E(λ 1 + λ 2 |λ 1 = x) is non-decreasing in x for x < 0.
Using the joint density of (λ 1 , λ 2 ) where Z is a Gaussian with mean µ and variance σ 2 truncated above zero. Then which is non-negative by the Cauchy-Schwarz inequality applied to Z 2 = Z 3/2 Z 1/2 . So in particular, E(λ 1 + λ 2 |λ 1 = x) is strictly increasing in x, as required. Using Gromov's theorem in the same way shows that in order to verify (5.7) for A of the form {λ 2 ≤ c 2 } or {λ 1 ≤ c 1 , λ 2 ≤ c 2 } it is enough to show that E(λ 1 + λ 2 |λ 2 = c 2 ) and E(λ 1 + λ 2 |λ 1 = c 1 , λ 2 ≤ d) are non-decreasing in c 2 > 0 and c 1 < 0 respectively. This can be proven using an identical calculation to that for d dx E(λ 2 |λ 1 = x) (the only change is the region on which Z is truncated), thus completing the proof of the lemma.

5.2.
Proof of Theorem 2.16. We now use Lemmas 5.5 and 5.7 to complete the proof of Theorem 2.16, treating the RPW case and the general case simultaneously.
We begin by fixing a realisation of g and θ. Let A( , R) denote the annulus on the plane centred at the origin with inner radius and outer radius R. We discretise this region by considering the points with polar coordinates for i, j = 1, . . . , 2 n . We consider these points as a graph by placing an edge between (r i 1 , ω j 1 ) and (r i 2 , ω j 2 ) if and only if |i 1 − i 2 | + |j 1 − j 2 | = 1. We define a site percolation model by declaring the vertex (r i , ω j ) open iff (r i , ω j ) − < 0 (so an edge is open precisely when both of its vertices are open). Let S ,R,n, denote the event that there is an open path between ( , θ) and ( , θ + π) in this percolation model. Let S ,R, denote the event that {f < }∩A( , R) contains a path joining ( , θ) to ( , θ +π). We claim that Sincef has no critical points at level away from the origin (Lemma 4.3), the level sets {f = } ∩ A( , R) consists of C 2+η curves. So in particular, if there exists a path in {f < }∩A( , R) joining ( , θ) to ( , θ +π), then for n sufficiently large we may assume this path lies on the graph with vertices (r j ) as defined above. Hence S ,R, ⊂ ∪ ∩ S ,R,n, . If there is no path in {f < } ∩ A( , R) joining ( , θ) to ( , θ + π) then there are three possibilities: (1)f − is non-negative at ( , θ) or ( , θ + π); (2) there exists a path in {f ≥ } ∩ A( , R) joining ( , ω i ) to ( , ω j ) for some ω i ∈ (θ, θ + π) and ω j ∈ (θ − π, θ), (here we assume that n is sufficiently large to find such ω i , ω j ); or (3) there exist two paths in {f ≥ } ∩ A( , R) which join ( , ω i ) and ( , ω j ) respectively to ∂B(R) for ω i , ω j as before (See Figure 8). In each of these cases, for n large enough we can construct corresponding paths on the discrete lattice as above which block a discrete path from joining ( , θ) to ( , θ + π) in {f < } and so S ,R,n, cannot occur for sufficiently large n. Therefore S ,R, ⊃ ∪ ∩ S ,R,n, , completing the proof of the claim.
Now let S R, be the event thatf has an R-lower connected saddle point at the origin. Conditional on θ, if this event occurs then so must S ,R, for sufficiently small. Conversely, if the saddle point at the origin is not R-lower connected, then it must be four arm in B(R) or R-upper connected. In either case S ,R, cannot occur for sufficiently small. We conclude that 1 S R, = lim →0 1 S ,R, and by applying the dominated convergence theorem to (5.12) we see that (5.13) P (S R, 1 |g, θ) ≤ P (S R, 2 |g, θ) . Figure 8. Two of the three ways in which S ,R, can fail, corresponding to cases (2) and (3) above.
Finally we let S be the event thatf has a lower connected saddle point at the origin and note that trivially S = ∪ R S R, . Applying this to (5.13) shows that P (S 1 |g, θ) ≤ P (S 2 |g, θ) . Integrating over realisations of g and θ implies that P(S 1 ) ≤ P(S 2 ) and so by definition (see the proof of Theorem 2.11) . A near identical argument shows that p * s + ( )/p s ( ) is non-increasing in . 5.3. Remaining results. We now prove the remaining results stated in Section 2.2., namely Corollaries 2.17 and 2.18 and Propositions 2.20-2.23.
Proof of Corollary 2.17. Since p * s − /p s is monotone it has at most a countable number of discontinuities, all of which are jump discontinuities. By the continuity of p s , the same is true of p * s − . Since c ES is absolutely continuous (Theorem 2.6) it is differentiable almost everywhere (see [26,Theorem 7.18]) with derivative p * s − − p m + . The density p m + is derived explicitly in [11] and is continuously differentiable. It also follows from monotonicity that p * s − /p s is differentiable almost everywhere, and since p s is smooth (again, from [11]) the same is true of p * s − , thus showing that c ES is twice differentiable almost everywhere. A similar proof applies to c LS .
Proof of Corollary 2.18. Since the equivalence of (2)-(4) follows from Theorem 2.6, and (1) implies (2) by Theorem 2.11, it remains to show that (2) implies (1). Now suppose there exists a version of p s − which is continuous on (a, b), denotedp s − . Thenp s − /p s = p * s − /p s almost everywhere, and since the former is continuous and the latter is monotone, this equality must hold pointwise on (a, b), so p * s − is continuous on (a, b). We note thatp s + := p s −p s − defines a continuous version of p s + and arguing as above then shows that p * s + is continuous on (a, b).
Therefore the almost everywhere equality p * s − + p * s + = p s is in fact true for all points in (a, b), and by (4.2)f almost surely has no infinite four arm saddle at the origin for all ∈ (a, b).
Proof of Proposition 2.23. We use the 'barrier method', that is, we show that the probability of having at least one component of {f ≥ } contained in B(r) is strictly positive for some fixed r > 0. By linearity of expectation and stationarity of f , this shows that lim inf R→∞ E(N ES (R, ))/R 2 > 0, so in particular c ES ( ) > 0.
It is known that the RPW has the orthogonal expansion where (r, θ) represents x in polar coordinates, J m is the m-th Bessel function and a m = b m + ic m = a −m with (b m ) m≥0 and (c m ) m∈N independent standard (real) Gaussians and c 0 = 0. Let r be the minimiser of J 0 , so r ≈ 3.83 and J 0 (r) < −0.4. We note that by considering the power series for the Bessel functions, it can be shown that for x ∈ [0, 4] and m ≥ 3, |J m (x)| ≤ (11/2)(2 n /n!). Finally we note that J m is bounded in absolute value by 1 for any m. Now consider the event that a 0 > min{| |, 1}, |a −1 |, |a −2 |, |a 1 |, |a 2 | ≤ C 1 and for |m| > 2, |a m | ≤ C 2 N !/4 n . It is easily seen that this event has positive probability, and for appropriately chosen constants C 1 , C 2 > 0, we see that on this event f (0) > and f (x) < for any x such that |x| = r. Therefore f has a component of {f ≥ } contained in B(r) with positive probability, completing the proof of the result.
Proof of Proposition 2.20. By Corollary 2.18 we may assume that p s − ( )/p s ( ) is non-decreasing.
In particular, p m + (x) = 0 for x < 0, so by Theorem 2.6 for < ≤ 0 Taking → −∞ shows that for < 0 By Proposition 2.23 this is positive for every < 0, so in particular there must exist arbitrarily negative x such that p s − (x) > 0. Since p s − ( )/p s ( ) is non-decreasing, we conclude that p s − is strictly positive for all ∈ R. Since p s − (x) = p s + (−x) we also see that p s + (x) > 0 for all x and since p s − + p s + = p s we see that 0 < p s − (x)/p s (x) < 1 for all x ∈ R. Finally, we note that there must exist a sequence n > 0 with n → 0 such that p s − ( n )/p s ( n ) ≥ 1/2 for all n. Indeed, if this were not true, by monotonicity, there would exist a neighbourhood of 0 on which p s − /p s < 1/2 and by symmetry p s + /p s < 1/2 on this neighbourhood, but then there would exist a set of positive measure on which p s − + p s + < p s giving a contradiction. Now let f be the BF field. By Theorem 2.16, p s − ( )/p s ( ) is non-decreasing and so for > 0 p s − ( ) p s ( ) ≥ p s− (0) p s (0) = 1 2 .
Therefore c ES ( ) ≥ p s ( )/2 − p m + ( ) for ≥ 0. Evaluating these densities explicitly then shows that this expression is strictly positive for ≤ 0.64.
Appendix A. Non-degeneracy We verify some claims about non-degeneracy of Gaussian fields: Lemma A.1. Let f be a C 2 stationary planar Gaussian field. Then the spectral measure µ being supported on two lines through the origin is equivalent to the Gaussian vector ∇ 2 f (0) being degenerate.
If ∇ 2 f (0) is degenerate, then we may choose a = 0 such that this expression is zero, and hence the integrand is identically zero on the support of µ. Hence the support of µ is contained in the zero set of a binary quadratic form which is contained in two lines through the origin.
Conversely if the support of µ is contained in the union of two lines through the origin, then we may choose a = 0 such that the zero set of a 1 x 2 1 + a 2 x 2 2 + a 3 x 1 x 2 is equal to this union. Hence the integral above will be zero and ∇ 2 f (0) will be degenerate.
By the same arguments as in the proof of Lemma A.1 If f does not satisfy Assumption 2.7, then there exists a choice of a such that this expectation is zero and one of the first three elements of a is non-zero. Hence the integrand above must be identically zero on the support of µ, and from the form of this integrand (and the fact that one of the first three elements of a is non-zero) we see that the support cannot contain an open interval of a non-degenerate conic section.