A G W Lawrie

M Duran-Matute

J F Scott

F Godeferd

J-B Flor

C Cambon

L Danaila

 offered a prediction of jet development, characterised solely by an unknown, constant entrainment coefficient. The presence of background rotation complicates the kinematics of the entrainment, and without special treatment, the jet suffers a helical instability. Here, we present one technique which stabilises the axisymmetric jet, yet preserves its desirable turbulent properties. The jet offers a steady-state flow in which there is an axial variation of local Rossby number, and after decay along the axis to a critical value, cones of inertial waves emerge. In this paper, we demonstrate these features using our numerical software MOBILE, offer our solution to stabilise the jet, and explain the mechanisms involved.

Introduction

Turbulence in rotating fluids has attracted the attention of many researchers since the experimental work of [START_REF] Hopfinger | Turbulence and waves in a rotating tank[END_REF] in which turbulence was created by continuously oscillating a grid by a small amplitude in a large tank. Rossby and Reynolds numbers decayed away from the grid due to viscous dissipation. However, critical questions arising from this work have remained unanswered for three decades, particularly regarding the interaction of turbulence and waves. More recent work by [START_REF] Davidson | On the evolution of eddies in a rapidly rotating system[END_REF] examined the problem from a different angle, choosing to study the emergence of structures from a region of decaying, rather than continuously forced turbulence. This has the obvious advantage that the turbulence can freely evolve for a few eddy-turnovers before the Rossby number is sufficiently low to generate wave-like structures, thus reducing the influence of spatial and temporal forcing of the grid. Our present work is a novel configuration which imposes neither spatial scales nor temporal forcing, maintains steady-state inhomogenous rotating turbulence, and has a spatially varying Rossby number.

In this paper, we consider an axisymmetric jet with its axis coincident with the axis of rotation of the whole system, and use cylindrical coordinates (e r , e θ , e z ), where z is the coordinate along the axis of the jet. In the case without background rotation, u jet ∼ 1 z and r jet ∼ z, as shown by [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF], and thus, the nominal jet Reynolds number is constant. Provided the structure of the jet remains unaffected by a background rotation, this would imply in the rotating case that the Rossby number Ro ∼ 1 z 2 . This combination of Re and Ro in a statistically stationary, unforced, and self-similar flow makes the jet a remarkably clean testcase for analysis of turbulence under the influence of the Coriolis force, since structural changes in the turbulence are, unlike grid generated turbulence for example, a function of one variable only. In contrast also to the plethora of work on swirling jets for applications to flame stabilisation, here, the influence of rotation increases (rather than decreases) with axial distance, and thus is robustly maintained long after the jet has 'forgotten' the effects of its emission from a finite aperture.

The following section describes the mechanism of instability we observe in the axisymmetric jet under the influence of rotation and presents an elegant solution to stabilise the jet, which we then compare with results from our MOBILE simulations.

Mechanisms

Azimuthal kinematics

Despite the immediate attractiveness of this testcase, a complication arises due to the jet entrainment. The entrainment flow at sufficiently large distance from the jet axis can be characterised as an axial sink with constant strength, assuming the axial variation of entrainment coefficient is small, and the large-scale flow in an infinite domain is approximately twodimensional. In the inertially-fixed reference frame, the planetary vorticity gives the initial circulation Γ 0 around each circumferential contour at radius r a finite value,

Γ 0 = (Ω × r).ds, (1)
where ds = rdθe θ is a contour element, Ω = [0, 0, Ω] is the rate of solid body rotation, and r is the position vector of a point on the contour. Kelvin's circulation theorem demands that the circulation on all material contours remains unchanged, ie.

DΓ Dt = 0, (2)
so as contours converge towards the axial sink, their radius reduces and it follows that the velocity component u.ds must increase to preserve the initial circulation. It is clear, therefore, that the far-field effect of the axial sink is amplification of the background vorticity, and that the far-field flow around our jet behaves rather similarly to the far-field flow induced by a tornado. We note that the near-field behaviour is not well-represented by an axial sink, and thus we do not require specific treatment of the singularity at the origin. Neglecting viscosity, the kinematics of the far-field are unbounded in time, since the circulation associated with each contour increases with radius, and contours at all radii eventually converge to the axis. Naturally we can neither numerically nor experimentally verify our analysis of the far-field in an unbounded domain; however, with our jet in a finite, enclosed domain, we observed azimuthal acceleration of the flow consistent with this idealised analysis. In a finite domain we also observe a helical instability; the jet initially grows, entrains ambient fluid which follows the far-field behaviour discussed above, and this helically displaces the jet from the axis. At large displacement amplitude, the jet breaks down, and the associated entrainment flow ceases. With nothing to drive further radial convergence of material contours, the azimuthal velocity decays sufficiently for the jet to reform, and a periodic formation/breakdown cycle is established.

The key development that lays the basis for our work on steady-state inhomogenous rotating turbulence, is the stabilisation of the far-field flow. Ultimately the problem can be recast as ensuring by some means that the circulation near the axis remains bounded in time. One simple way to achieve this in a finite domain is to enforce radial inflow (ie. no azimuthal velocity) at some distance r dom from the jet, thus limiting the maximum circulation to |Ω × r dom | .ds, and satisfying continuity by evacuating fluid through a plane at suitably large z. While appropriate for a finite domain, as one requires in an experiment, there are obvious theoretical and practical limitations to this approach. Instead we focus on a much more elegant solution.

We begin by recognising that in the spatially unbounded problem, contours converge from infinity, and we obtain persistent azimuthal acceleration over time. If one were able to modify the problem such that contours converge from a constant and finite radius, one would recover a steady-state with finite circulation near the axis. We achieve this transformation by adding a small axial velocity u a to the ambient fluid, rather like a co-axial jet with an outer flow of infinite radius. In the inertially fixed reference frame, flow emanating from the boundary plane has velocity,

u ambient = Ωre θ + u a e z , (3)
thus material contours emerge from a bottom boundary plane in solid body rotation, and converge towards the axis in space. Contours of small initial radius reach the axis at low values of z and have relatively little circulation, those with larger initial radii reach the axis at larger values of z with greater circulation. The key to this spatio-temporal transformation is that everywhere except for the contour at infinite initial radius, the azimuthal velocity obtained near the axis remains finite and invariant with time, though increasing with z. It is perhaps clearer in the rotating frame to see why the velocities are bounded. If the relative circulation Γ ′ is defined to be zero in solid body rotation, then by considering the projection onto a plane with the z-axis as its normal, we can use Stokes' theorem to transform the line integral around the contour into a double integral across the enclosed, projected area, A. By projecting the relative velocity field onto the same plane, we obtain an evolution equation for the relative circulation,

DΓ ′ Dt = - 2Ω∇ 2D .u ′ dA, (4)
where ∇ 2D operates solely on the in-plane relative velocity components. Here, one can see that the relative circulation of a material contour is determined by the time-integral of the sink strength within the contour for the duration of travel of the contour from initial to final radius. Thus, if material contours emanate from a boundary at finite radius, their azimuthal velocity remains bounded.

2.2. Kinematics in the (r, z) plane By considering continuity along individual stream-tubes, the above arguments quite clearly generalises to the case of axially varying sink strength, which corresponds to the situation one might expect from axially developing turbulent anisotropy. The streamline behaviour outside the jet arises from the balance of entrainment and axial co-flow, and since the azimuthal velocity u θ is perpendicular to the (r, z) plane and independent of θ, we expect the u r and u z components of velocity to be approximately irrotational in the interior. By construction the co-flow imposes a streamfunction boundary condition Ψ(r, 0) ∼ r 2 , and modelling the jet entrainment as a line sink corresponds to Ψ(r * , z) ∼ z, where r * represents the edge of the jet. Since we require axial co-flow, it follows that the streamfunction is not analytic at (r * , 0), and to resolve this we need to add a point vortex, analogous to a Kutta condition on the boundary. The lateral far-field boundary is somewhat simpler; the streamline at r = ∞ takes infinitely long to reach the axis, and thus is vertical with Ψ = const.. The schematic in figure 1 illustrates how one might expect the far-field streamline pattern to appear, and shows the contour at infinity as a dashed line.

Jet dynamics in the high Rossby number region

The volume and axial momentum conservation arguments originally proposed by [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF]) also appear to hold in the transformed case, since one might assume that the local entrainment is not affected by the shape of the entrained stream-tube in the far-field. This is a reasonable assumption provided the characteristic eddy velocity is significantly faster than the azimuthal velocity of the entrained fluid at the edge of the jet. By conserving circulation it is trivial to show that the azimuthal velocity of a material contour originating at radius r is given by

u θ = Ωr 2 r * , (5)
where r * is its current radius. If r * is taken as the edge of the jet, r * ∼ z, and noting that all volume emanating from the bottom boundary eventually reaches the axis, r ∼ z 2 , then

u θ ∼ z 3 . ( 6
)
There exists therefore a region in which our original assumption is valid, sufficiently far from the origin (we require that the jet commences with a finite aperture to circumvent the singularity at the axis), but before u jet -u a ∼ O(u θ ).

Jet dynamics in the low Rossby number region

Once the Rossby number of the jet approaches order unity, both the core jet flow and the relatively slow entrained flow are influenced by rotation. It is well-known that inertial waves propagate from eddies in a rotating ambient, though the mechanism of generation has been the focus of widespread debate in recent years. In the rotating reference frame, one obtains the form of a linear wave equation by neglecting the nonlinear u.∇u term in the momentum equation, ie.

∂ 2 û ∂t 2 + ω2 û = 0, ( 7
)
where û is the relative velocity field Fourier-transformed onto a set of basis vectors k, and ω has dimensions of 1 T . The dispersion relation,

ω = ± 2Ω.k |k| , (8)
is unusual, since surfaces of constant phase ripple in the direction of the phase velocity,

c p = ωk k.k = ± 2(Ω.k)k |k| 3 , ( 9
)
while energy is transported in the direction of the group velocity,

c g = ∇ω(k) = ± 2k × (Ω × k) |k| 3 , (10)
and it can be shown that the two directions are perpendicular. As demonstrated very beautifully in [START_REF] Godeferd | Direct numerical simulations of turbulence with confinement and rotation[END_REF], inertial waves propagate in axisymmetric cones of constant angle to the rotation axis, with equatorial wave-vectors k corresponding to axial propagation, and axial wave-vectors being the limiting case of slow, radially propagating inertial waves. The axisymmetry of our jet configuration is thus a very powerful feature, since it corresponds directly to the physical nature of the inertial waves. In homogenous turbulence, one can conceive of a sea of inertial waves of all directions, with, on average, all incoming and outgoing inertial waves destructively interfering. This elegant symmetry is unavailable in inhomogenous cases such as our jet. Waves that propagate with group velocities at angles close to the rotation axis remain within the jet and interact with the turbulence, but those that radiate more horizontally leave the jet, remove energy, and act as an angular filter on wave-turbulence interactions.

Results

Computational method

Following the Large Eddy Simulation work of Lawrie & Dalziel (2011a,b), for our preliminary calculations presented here, we use a third order, staggered grid, fractional step methodology to solve the following equations of motion in the relative frame,

∂(ρu ′ i ) ∂t + ∂ρu ′ i u ′ j ∂x j = - ∂p ′ ∂x i + 2ǫ ijk u ′ j Ω k (11) subject to the constraint ∂u ′ i ∂x i = 0, ( 12
)
which is enforced exactly through a pressure re-distribution at each timestep. A parallel multigrid scheme is used to accelerate the pressure convergence. We use stress-free zero flux conditions on the side-walls of the domain to mimic the pressure field of our computational setup while neglecting the influence of thin boundary layers on the walls, and an inflow boundary condition that represents both the unswirled ambient inflow and the nozzle aperture. The outflow is a balanced flux, stress-free condition approximately transmissive to inertial waves with a spatial frequency well-resolved on the mesh. In the calculation we present here, the mesh was formed from cubic cells, 256 × 256 × 1024, with the large direction aligned with the nozzle and the rotation axis. Even with this size of calculation, the nozzle aperture occupies only O(10) cells of the boundary surface, and hence this region is inadequately resolved. However, in the critical Rossby number regions that are of specific interest in this work, the inertial waves, the integral scale of the turbulence and perhaps a decade of inertial range are well resolved.

Discussion

The numerical work we present here is part of a larger collaborative effort to study the evolution of anisotropic turbulence, and the stabilised jet configuration is the focus of the numerical and experimental components thereof. We show with our preliminary calculations firstly that a quasi-steady state can indeed be achieved for the co-flowing jet, and secondly that our key scalings are well-matched by the data. In figure 2, which renders radial velocity in a thin volume around the jet axis, by comparing the plots taken at times τ Ω = 3 and τ Ω = 6, where τ Ω = t 2πΩ . The other parametric timescale in the problem is the axial advection time-scale τ z = z dom ua , and for the simulation shown the corresponding values are τ z = 1.25 and τ z = 2.5. The testcase selected here uses u j ua = 4 and for this relatively large co-flow velocity ratio, the velocity field does not display the strong helical instability of the zero co-flow case, despite there being a strong and vertically increasing azimuthal velocity. The spread angle of the jet, with respect to the classical [START_REF] Morton | Turbulent gravitational convection from maintained and instantaneous sources[END_REF] model, is reduced by the co-flow. In common with non-rotating co-flowing jets, the temporal rate of increase in jet radius is maintained by entrainment, but the axial advection reduces the spatial rate of growth, leading to its distinctive, shallow angle form. We performed this calculation in a large-aspect-ratio domain, to ensure that we captured the important jet development regimes, and that we can quantify deviation from our scalings due to the confinement of the computational domain.

In an unconfined domain, material contours approaching the axis at continuously increasing z emanate from continuously increasing r, but this is no longer possible when there is a radial limit to the domain, and thus r dom sets the maximum circulation of any entrained contour, and hence bounds u θ . In cases where the volume flux that it is possible for an unconfined jet to entrain is a significant proportion of the co-flow volume flux, we expect a laterally confined case to exhibit reduced entrainent. In the calculation presented in figure 2 we chose u a such that the volume flux entrained into the jet is weak relative to the co-flow volume flux, so accordingly we expect negligible adjustment of the entrainment. Figure 3 shows the actual azimuthal velocity, and there are several points of interest to be noted here. The resolution of the flow in the vicinity of the nozzle is too poorly resolved to allow turbulence to develop immediately in a natural manner, and consequently the entrainment flow cannot develop either. The virtual origin that arises is located at z = 0.2, and one can consider this as being analogous to the location of a nozzle aperture. Interestingly, in this situation where axial co-flow is only enforced some distance behind the 'nozzle', one obtains a vertically inhomogeneous axial sink. Thus streamlines in (r, z) emanating from the bottom boundary must converge towards the bottom extremity of the sink, and this induces anti-cyclonic azimuthal velocity (vortex compression). Once the entrainment flow begins, at z > 0.2, ∂u θ ∂z becomes positive and follows the z 3 prediction closely (vortex stretching). At sufficiently large z, it becomes aware of the finite lateral boundary, and u θ ceases to increase monotonically.

The most valuable feature that we encounter is the Rossby number variation with axial distance. While this does occur in swirling jets (used for instance to stabilise flames), there, the influence of rotation decays rapidly as the initial circulation is diffused radially by turbulent transport, and so low Rossby numbers are restricted to a region close to the nozzle, and it becomes difficult to separate effects caused by a finite aperture nozzle from those caused by rotation. In our novel case, however, the jet approaches a self-similar form some distance downstream of the nozzle, but with appropriate selection of rotation rates, this occurs well before the Rossby number falls below a critical value. As noted earlier, substantial under-resolution of turbulent scales close to the nozzle leads to a large displacement of the virtual origin, and here, such separation of length-scales is particularly desirable. Figure 4 shows the Rossby number as a function of height, measured from the turbulent integral length-and velocity-scales,

Ro turb = u turb Ωl turb , (13)
and it follows reassuringly closely the scaling we predicted in the introduction by extrapolating from the non-rotating non-co-flowing jet. Thus we can examine the transition from turbulent isotropy to anisotropy simply by reading off time-averaged statistics at each axial position.

Conclusions

In this paper we have presented, for the first time to our knowledge, a means of obtaining unforced statistically stationary rotating turbulence together with a means of varying Rossby number in space rather than in time in a spatially self-similar flow. Our approach uses a free jet subject to background rotation and while the most obvious configuration leads to unbounded azimuthal acceleration, a small modification mitigates this effect and thus the concept has practical utility. As part of a larger collaborative study on anisotropic turbulence, combining theoretical, numerical and experimental tools, we present our initial findings from our numerical software MOBILE on our novel jet configuration.

Figure 1 .

 1 Figure 1. Far-field streamline pattern, modelling the jet as an axial sink.

Figure 2 .

 2 Figure 2. False-colour perspective rendering of radial velocity in a thin volume at t 2πΩ = (left) 3, (middle) 6, and (right) a close-up of (middle) in the low Rossby number region, detailing eddy elongation by inertial waves towards the edge of the jet.

Figure 3 .

 3 Figure 3. Azimuthal velocity averaged over planar annuli 0.35r dom > r > 0.3r dom > r jet as a function of height, normalised by Ωr 2 dom /r. The location of the virtual origin is shown by a dashed line.

Figure 4 .

 4 Figure 4. Rossby number as a function of height, with Ro ∼ 1 z 2 scaling superimposed.

Acknowledgments

The authors would like to thank the French National Research Agency, ANR for their financial support through the project ANISO for this collaboration.


N Chalmoukis
email: nikolaos.chalmoukis2@unibo.it

A Hartmann
email: andreas.hartmann@math.u-bordeaux.fr

K Kellay
email: karim.kellay@math.u-bordeaux.fr

B D Wick
email: wick@math.wustl.edu

RANDOM INTERPOLATING SEQUENCES IN DIRICHLET SPACES

Keywords: 2010 Mathematics Subject Classification. 30D05, 30E05, 30B20, 31C25 Interpolating sequences, separation, Carleson measure, random sequences

We discuss random interpolation in weighted Dirichlet spaces D α , 0 ≤ α ≤ 1. While conditions for deterministic interpolation in these spaces depend on capacities which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at α = 1/2 in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for D α are exactly the almost sure separated sequences when 0 ≤ α < 1/2 (which includes the Hardy space H 2 = D 0 ), and they are exactly the almost sure zero sequences for D α when 1/2 ≤ α ≤ 1 (which includes the classical Dirichlet space D = D 1 ).

Introduction

Understanding interpolating sequences is an important problem in complex analysis in one and several variables. The characterization of when a sequence of points is an interpolating sequence finds many applications to different problems in signal theory, control theory, operator theory, etc. In classical spaces like Hardy, Fock and Bergman spaces, interpolating sequences are now well understood objects, at least in one variable [START_REF] Garnett | Bounded analytic functions[END_REF][START_REF] Seip | Interpolation and sampling in spaces of analytic functions[END_REF][START_REF] Zhu | Analysis on Fock spaces[END_REF]. In Dirichlet spaces, it turns out however that getting an exploitable description of such interpolating sequences is a notoriously difficult problem related to capacities. Crucial work has been undertaken in the 90s by Bishop and Marshall-Sundberg (see more precise indications below). However, while easier checkable sufficient conditions were given by Seip in the meantime (see [START_REF] Seip | Interpolation and sampling in spaces of analytic functions[END_REF]), no real progress in the understanding of these sequences has been made since those works. In such a situation, a probabilistic approach can lead to a new vision of these interpolating sequences. Note that besides the Hardy and Bergman spaces, the Dirichlet space, and its weighted companions, are among the most prominent spaces of analytic functions on the unit disk. They appear naturally in problems on classical function theory, potential theory, as well as in operator theory when one investigates for instance weighted shifts.

Here we consider random sequences of the following kind. Let Λ(ω) = {λ n } with λ n = ρ n e iθn (ω) where θ n (ω) is a sequence of independent random variables, all uniformly distributed on [0, 2π] (Steinhaus sequence), and ρ n ∈ [0, 1) is a sequence of a priori fixed radii. Depending on distribution conditions on (ρ n ) as will be discussed below, we ask about the probability that Λ(ω) is interpolating for Dirichlet spaces D α , 0 ≤ α ≤ 1. Recall that the weighted Dirichlet space D α , 0 ≤ α ≤ 1, is the space of all analytic function f on the unit disc D such that

f 2 α := |f (0)| 2 + D |f (z)| 2 (1 -|z| 2 ) 1-α dA(z) < ∞,
where dA(z) = dxdy/π stands for the normalized area measure on D (we refer to [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF] for Dirichlet spaces). For f (z) = n≥0 a n z n , z ∈ D, the above expression is equivalent to n≥0 (1 + n) α |a n | 2 . If α = 0, D 0 is the Hardy space H 2 , and the classical Dirichlet space D corresponds to α = 1.

Recall that in a Hilbert space H of functions analytic in the unit disk D equipped with a reproducing kernel k λ , i.e. f (λ) = f, k λ H for every λ ∈ D and f ∈ H (a so-called reproducing kernel Hilbert space), a sequence Λ of distinct points in D is called (universal) interpolating if {(f (λ)/ k λ H ) λ∈Λ : f ∈ H} = 2 (for the difference between interpolating and universal interpolating sequences see below). Concerning the deterministic case of interpolation in the classical Dirichlet space D, in unpublished work Bishop [START_REF] Bishop | Interpolating sequences for the Dirichlet space and its multipliers[END_REF] and, independently, Marshall-Sundberg [START_REF] Marshall | Interpolating sequences for the multipliers of the Dirichlet space[END_REF] characterized the interpolating sequences. The first published proof was given by Bøe [START_REF] Bøe | A norm on the holomorphic Besov space[END_REF] who provides a unifying scheme that applies to spaces that satisfy a certain property related to the so-called Pick property (see [START_REF] Agler | Pick interpolation and Hilbert function spaces[END_REF][START_REF] Seip | Interpolation and sampling in spaces of analytic functions[END_REF]), and Dirichlet spaces fall in this category. For these spaces Λ is a (universal) interpolating sequence if and only if Λ is H-separated (i.e sup λ =λ * ∈Λ | k λ / k λ H , k λ * / k λ * H | < δ Λ < 1) and

µ Λ = λ∈Λ δ λ / k λ 2 H is a Carleson measure for H (i.e, D |f | 2 dµ ≤ C Λ f 2 H ).
Recently, Aleman, Hartz, McCarthy and Richter [START_REF] Aleman | Interpolating Sequences in Spaces with the Complete Pick Property[END_REF] have shown that this characterization remains valid in arbitrary reproducing kernel Hilbert spaces satisfying the complete Pick property. Stegenga [START_REF] Stegenga | Multipliers of the Dirichlet space[END_REF] characterized Carleson measures for Dirichlet spaces, but his characterization is based on capacities which are notoriously difficult to estimate for arbitrary unions of intervals. There are other characterizations of Carleson measures in Dirichlet spaces, see [START_REF] Arcozzi | Carleson measures for analytic Besov spaces[END_REF][START_REF] Arcozzi | Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls[END_REF], as well as [START_REF] Arcozzi | Dirichlet Spaces and Related Spaces[END_REF][START_REF] El-Fallah | A primer on the Dirichlet space[END_REF] and references therein, but which are not easily interpreted geometrically for interpolating sequences. Finally, we mention related work by Cohn [START_REF] Cohn | Interpolation and multipliers on Besov and Sobolev spaces[END_REF] based on multipliers.

The interesting feature in connection with Stegenga's result is that switching to the random setting, it turns out that his condition for unions of intervals reduces to a one-box condition involving just one interval, and for which the capacity can be estimated. This will be of central relevance for our discussions.

We also would like to observe that generally, when the deterministic frame does not give a full answer to a problem, or if the deterministic conditions are not so easy to check, it is interesting to look at the random situation. In particular, it is interesting to ask for conditions ensuring that a sequence picked at random is or is not interpolating almost surely (i.e., which are in a sense "generic situations"?). In this context, it is also worth mentioning the huge existing literature around Gaussian analytic functions which investigates the zero distribution in classes of such functions [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF].

The problems we would like to study in this paper are inspired by results by Cochran [START_REF] Cochran | Random Blaschke products[END_REF] and Rudowicz [START_REF] Rudowicz | Random sequences interpolating with probability one[END_REF] who considered random interpolation in the Hardy space. Since interpolation in this space is characterized by separation (in the pseudohyperbolic metric) and by the Carleson measure condition (note that the Hardy space was the pioneering space with a kernel satisfying the complete Pick property), those authors were interested in a 0-1 law for separation, see [START_REF] Cochran | Random Blaschke products[END_REF], and a condition for being almost surely a Carleson measure [START_REF] Rudowicz | Random sequences interpolating with probability one[END_REF], which led to a 0-1 law for interpolation. It is thus natural to discuss separation, Carleson measure type conditions and interpolation in Dirichlet spaces.

Concerning separation in Dirichlet spaces D α , 0 < α < 1, this turns out to be the same as in the Hardy spaces (see [26, p.22]), so that in that case Cochran's result perfectly characterizes the situation. The separation in the classical Dirichlet space, however, is much more delicate than in the Hardy space. We establish here a 0-1type law for separation in D. While our proof of this fact is inspired by Cochran's ideas, it requires a careful adaptation to the metric in that space.

Concerning Carleson measure type results in Dirichlet spaces, 0 < α < 1, it is instructive to first rediscuss the situation in the Hardy space. As it turns out we are able to improve Rudowicz' result with a simplified proof based on generating functions and Markov's inequality instead of Rudowicz's proof using Tchebychev's inequality. This proof carries over to the Dirichlet situation and allows, together with Stegenga's characterization of Carleson measures, to discuss the results on interpolation in D α . A peculiar breakpoint in the behaviour of such interpolating sequences depending on the weight α is now appearing: for 0 ≤ α < 1/2, almost sure separation corresponds to almost sure interpolation, while for 1/2 ≤ α ≤ 1, almost sure zero sequences correspond to almost sure interpolating sequences. Observe that for the critical value α = 1/2 the behaviour is the same as in the Dirichlet space. Commenting further on this breakpoint α = 1/2, we would like to mention work by Newman and Shapiro [START_REF] Newman | The Taylor coefficients of inner functions[END_REF] who show that if I is a non-constant singular inner function, then

I / ∈ D 1/2 .
Since zero sequences are of some importance as we have just seen, another central ingredient of our discussion is a rather immediate adaption of Bogdan's result on almost sure zero sequences in the Dirichlet space to the case of weighted Dirichlet spaces which we add for completeness in an annex.

As usual, the definition of interpolating sequences is based on the reproducing kernel of D α :

(1.1) k z (w) =      1 zw log 1 1 -zw if α = 1, 1 (1 -zw) 1-α , if 0 ≤ α < 1.
We mention that contrary to the Hardy space situation, it turns out that in certain spaces (e.g. the Dirichlet space) there exist two notions of interpolation depending on whether the restriction operator

R Λ : H -→ 2 , R Λ f = (f (λ)/ k λ H ) λ∈Λ takes values in 2 or not. Notice that R Λ f 2 2 = D |f | 2 dµ Λ . In D α , 0 ≤ α ≤ 1, a sequence Λ of distinct points in D said to be • a (simple) interpolating sequence if R Λ : f → (f (λ)/ k λ α ) λ∈Λ is onto 2 ,
i.e. the interpolation problem f (λ) = a λ has a solution f ∈ D α for every sequence (a λ ) with (a λ / k λ α ) λ∈Λ ∈ 2 (Λ), • a universal interpolating sequence if it is interpolating and moreover R Λ is well defined from D α into 2 .

Sequences which are interpolating for the Dirichlet space but not universally interpolating were discovered by Bishop in [START_REF] Bishop | Interpolating sequences for the Dirichlet space and its multipliers[END_REF], and they were further analyzed in [START_REF] Arcozzi | Onto Interpolating Sequences for the Dirichlet Space[END_REF] and [START_REF] Chalmoukis | Onto Interpolation for the Dirichlet Space and for[END_REF].

In this paper we will exclusively consider universal interpolation. In particular, when speaking about interpolation we mean universal interpolation.

We will now discuss in details the results we have obtained in this paper.

1.1. Back to the Hardy space. As pointed out in the introduction, before considering the situation in the Dirichlet space, it seems appropriate to re-examine the situation in the Hardy space. Recall that Cochran established a 0-1 law for (pseudohyperbolic) separation (see Theorem 1.3 below) and Rudowicz showed that Cochran's condition for separation implies almost surely the Carleson measure condition. This implies that interpolation is characterized by the condition ensuring almost sure separation. In Dirichlet spaces, however, the situation is quite different. So, in order to get a better understanding of the situation we start stating an improvement of Rudowicz' results on random Carleson measures in the Hardy space which will help to better understand the case of Dirichlet spaces.

Recall that the measure dµ Λ = λ∈Λ (1 -|λ| 2 )δ λ is called a Carleson measure for H 2 if there is a constant C such that for every interval I ⊂ T,

µ Λ (S I ) ≤ C|I|,
where |E| denotes normalized Lebesgue measure on T of a measurable set E, and

S I = z = re it ∈ D : e it ∈ I, 1 -|I| ≤ r < 1
is the usual Carleson window (see [START_REF] Garnett | Bounded analytic functions[END_REF]). We will prove that a weaker condition than Rudowicz' leads to Carleson measures almost surely in the Hardy space. We first need to introduce a notation: let

A n = {λ ∈ Λ(ω) : 1 -2 -n ≤ |λ| < 1 -2 -n-1 }, n = 0, 1, 2 . . .
be a dydadic annulus. Then we denote by N n , the number of points of Λ contained in A n :

N n = #A n . Theorem 1.1. Let β > 1 and suppose n≥1 2 -n N β n < +∞.
Then the measure dµ Λ is a Carleson measure almost surely in the Hardy space.

As a result, the Carleson measure condition alone is not sufficient to give a 0-1 law for interpolation in the Hardy space.

Note that Rudowicz [START_REF] Rudowicz | Random sequences interpolating with probability one[END_REF] showed that the above condition with β = 2 is sufficient. We will construct an example showing that it is not possible to replace β by 1, so that (almost sure or not) zero sequences do not imply almost surely the Carleson measure condition. This makes the Hardy space a singular point in this respect within the scale of weighted Dirichlet spaces, 0 ≤ α ≤ 1.

1.2. Interpolation in Dirichlet spaces D α , 0 < α < 1. In this subsection we mention the results connected to interpolation: zero sequences, separation, and Carleson measures. 1.2.1. Random zero sequences in Dirichlet spaces. A central role in our interpolation results will be played by random zero sequences. Indeed, for an interpolating sequence in the Dirichlet space it is necessary to be a zero sequence (interpolation implies that there are functions vanishing on the whole sequence except for one point λ, and multiplying this function by (z -λ) yields a nontrivial function in the Dirichlet space vanishing on the whole sequence). We recall some results on random zero set in Dirichlet spaces. Carleson proved in [START_REF] Carleson | On a class of meromorphic functions and its associated exceptional sets[END_REF] that when

(1.2) λ∈Λ k λ -2 α < ∞
then the Blaschke product B associated to Λ belongs to D α , 0 < α < 1 (for α = 0 this corresponds to the Blaschke condition for the Hardy space). When α = 1, Shapiro-Shields proved in [START_REF] Shapiro | On the zeros of functions with finite Dirichlet integral and some related function spaces[END_REF] that the condition (1.2) is sufficient for {λ} λ∈Λ to be a zero set for the classical Dirichlet space D 1 , see also [START_REF] Seip | Interpolation and sampling in spaces of analytic functions[END_REF]Theorem 1]

. Note that if 0 ≤ α < 1 then λ∈Λ k λ -2 α λ∈Λ (1 -|λ|) 1-α n 2 -(1-α)n N n and if α = 1 then λ∈Λ k λ -2 1 λ∈Λ | log(1 -|λ|)| -1 n n -1 N n .
On the other hand, it was proved by Nagel-Rudin-Shapiro in [START_REF] Nagel | Tangential boundary behavior of functions in Dirichlettype spaces[END_REF] that if {r n } ⊂ (0, 1) does not satisfy (1.2), then there is {θ n } such that {r n e iθn } is not a zero set for D α . Bogdan [START_REF] Bøe | A norm on the holomorphic Besov space[END_REF]Theorem 2] gives a condition on the radii |λ n | for the sequence Λ(ω) to be almost surely zeros sequence for D:

(1.3) P (Λ(ω) is a zero set for D) = 1 0 if and only if n n -1 N n < ∞ = ∞.
Bogdan's arguments carry over to D α , α ∈ (0, 1). For the sake of completeness, we will prove in the annex, Section 6, the following result on almost sure zero sequences.

Theorem 1.2. Let 0 ≤ α < 1. Then (1.4) P (Λ(ω) is a zero set for D α ) = 1 0 if and only if n 2 -(1-α)n N n < ∞ = ∞. 1.2.2. Interpolation in Dirichlet spaces D α , 0 < α < 1.
As pointed out earlier, interpolation is intimately related with separation conditions and Carleson measure type conditions. Recall that a sequence Λ is called (pseudohyperbolically

) separated if inf λ,λ * ∈Λ λ =λ * ρ(λ, λ * ) = inf λ,λ * ∈Λ λ =λ * |λ -λ * | |1 -λλ * | ≥ δ Λ > 0.
Since in Dirichlet spaces D α , 0 ≤ α < 1, the natural separation (D α -separation) is indeed pseudohyperbolic separation [26, p.22], we recall Cochran's separation result on pseudohyperbolic separation. Theorem 1.3 (Cochran). A sequence Λ(ω) is almost surely (pseudohyperbolically) separated if and only if

(1.5) n 2 -n N 2 n < +∞.
We should pause here to make a crucial observation. We have already mentioned that interpolating sequences are necessarily zero-sequences. Also separation is another necessary condition for interpolation. Now the condition for zero sequences (1.4) depends on α while the separation condition does not, and it follows that depending on α, it is one condition or the other which is dominating. From (1.4) and (1.5) it is not difficult to see that this breakpoint is exactly at α = 1/2 (for α = 1/2, (1.4) still implies (1.5)). This motivates already the necessary conditions of our central Theorem 1.5 below. Another ingredient of that result comes from Carleson measures which have been characterized by Stegenga using capacitory conditions. For these we have the following result which is in the spirit of Theorem 1.1 in the Hardy space.

Theorem 1.4. Let 0 < α < 1. If (1.6) n 2 -(1-α)n N n < ∞ then µ Λ = λ∈Λ (1 -|λ| 2 ) 1-α δ λ is almost surely a Carleson measure for D α .
Observe that contrarily to the Hardy space, where we had to pick β > 1, here the exponent is exactly (1 -α). We obtain a similar result in the classical Dirichlet space. So, in view of Theorem 1.2, we can deduce that almost sure zero sequences give rise to almost sure Carleson measures in D α , 0 < α ≤ 1 (which in particuliar includes the classical Dirichlet space).

We can now state our first main result.

Theorem 1.5.

(i) Let 0 < α < 1/2. Then P Λ(ω) is interpolating for D α = 1 0 if and only if n 2 -n N 2 n < ∞ = ∞. (ii) Let 1/2 ≤ α < 1. Then P Λ(ω) is interpolating for D α = 1 0 if and only if n 2 -(1-α)n N n < ∞ = ∞.
An interesting reformulation of the above results connects random interpolation with random zero sequences and random separated sequences as stated in the following corollary.

Corollary 1.6. The following statements hold:

(1) Observe that for α = 1/2, the condition for being almost surely a zero sequence is strictly stronger than the condition for being almost surely separated so that in the limit case α = 1/2 it is indeed the almost sure zero condition that drives the situation.

Let 0 ≤ α < 1/2.
1.3. Interpolation in the classical Dirichlet space. For the classical Dirichlet space we will first establish a result on separation, and then use again the fact that in the random situation Stegengas capacitory condition on unions of intervals reduces to a single interval.

1.3.1. Separation in the Dirichlet space. In the case α = 1, the separation is given in a different way. Let

ρ D (z, w) = 1 - |k w (z)| 2 k z (z)k w (w) , z, w ∈ D. A sequence Λ is called D-separated if inf λ,λ * ∈Λ λ =λ * ρ D (λ, λ * ) > δ Λ > 0
for some δ Λ < 1. This is equivalent to (see [26, p.23])

(1.7) (1 -|λ| 2 )(1 -|λ * | 2 ) |1 -λλ * | 2 ≤ (1 -|λ| 2 ) δ 2 Λ , λ, λ * ∈ Λ.
For separation in the Dirichlet space D we obtain the following 0-1 law.

Theorem 1.7.

P (Λ(ω) is D-separated) =      1, if ∃γ ∈ (1/2, 1) such that n 2 -γn N 2 n < ∞, 0, if ∀γ ∈ (1/2, 1) we have n 2 -γn N 2 n = ∞.
We observe that in both conditions we can replace the sum by a supremum (this amounts to replacing γ by a slightly bigger or smaller value). The lower bound 1/2 for γ is not very important, since it is the behavior close to the value 1 which counts.

As we will see, this is a very mild condition in comparison to the almost sure zero-or Carleson-measure condition, and does not play a big role for the almost sure interpolation problem. 1.3.2. Interpolation in the Dirichlet space D. Recall that Bogdan showed that Λ(ω) is almost surely a zero sequence for D if and only if n n -1 N n < +∞. This motivates already the necessary part of the following complete characterization of almost surely universal interpolating sequences for D. The sufficiency comes essentially from the fact that the condition easily implies the (mild) separation condition, as well as the Carleson measure condition which will be shown in a similar fashion as in Theorem 1.4. Theorem 1.8.

P Λ(ω) is universal interpolating for D = 1 0 if and only if n n -1 N n < ∞ = ∞.
We can reformulate the above result in the same spirit as Corollary 1.6

Corollary 1.9. A sequence is almost surely interpolating for D if and only if it is almost surely a zero sequence for D.

1.4. Organization of the paper. This paper is organized as follows. In the next section we present the improved version of the Rudowicz result concerning random Carleson measures in Hardy spaces which is the guideline for the corresponding result in the Dirichlet space. Indeed, this largely clarifies and simplifies not only the situation in the Hardy space, but also indicates the direction of investigation for the Dirichlet space. This will allow us to start Section 3 with the discussion of the Carleson measure result in D α , 0 < α < 1. We then solve completely the interpolation situation in these spaces. The next section is devoted to the 0-1 law on separation in the classical Dirichlet space. This requires a subtle adaption of the Cochran discussion in the Hardy space to the much more intricate geometry in the Dirichlet space. The proofs of the results on interpolating sequences in the classical Dirichlet space are contained in Section 5. Actually, as in the Hardy space, the core of the proof being probabilistic, we are able to get rid of analytic functions.

In the final Section 6, we give some indications to the 0-1 law on zero-sequences in weighted Dirichlet spaces based on Bogdan's proof in the classical Dirichlet space.

A word on notation. Suppose A and B are strictly positive expressions. We will write A B meaning that A ≤ cB for some positive constant c not depending on the parameters behind A and B. By A B we mean A B and B A. We further use the notation A ∼ B provided the quotient A/B → 1 when passing to the suitable limit.

Carleson condition in the Hardy space

Before considering Carleson measure conditions in the Dirichlet space, we will discuss the situation in the Hardy space, in particular we will prove here Theorem 1.1. We will also construct an example showing that it is not possible to choose β = 1 in the statement of Theorem 1.1.

2.1.

Proof of Theorem 1.1. We start introducing some notation. Let

I n,k = {e 2πit : t ∈ [k2 -n , (k + 1)2 -n )} n ∈ N, k = 0, 1, . . . ,
µ Λ (S n,k ) = m≥n 2 -m X n,m,k ≤ C2 -n
almost surely, in other words we have to prove

sup n,k 2 n m≥n 2 -m X n,m,k ≤ C
almost surely (in ω). The estimate above had already been investigated by Rudowicz [START_REF] Rudowicz | Random sequences interpolating with probability one[END_REF]. Here we will proceed in a different way with respect to Rudowicz' argument to obtain an improved version of his result and which allows to better understand the Dirichlet space situation.

Proof of Theorem 1.1. In view of our preliminary remarks, we need to look at the random variable

Y n,k = 2 n +∞ m=n 2 -m X n,m,k ,
where, as said above, X n,m,k ∼ B(2 -n , N m ). Hence, saying that Y n,k ≥ A means that there are Carleson windows for which the Carleson measure constant is at least A. Also denote by G Y n,k the probability generating function of the random variable Y n,k , i.e. G Y n,k (s) = E(s Y n,k ). It is well known that for a random variable X which follows a binomial distribution with parameters p, N we have that G X (s) = (1 -p + ps) N .

By the hypothesis, for n sufficiently large, N n ≤ 2 (1-)n , = 1 -1/β. Introduce now two parameters A, s > 0 to be specified later. By Markov's inequality we have

that log P (Y n,k ≥ A) = log P (s Y n,k ≥ s A ) ≤ log 1 s A G Y n,k (s) = m≥n N m log(1 -2 -n + 2 -n s 2 n-m ) -A log(s) ≤ 2 -n m≥n N m (s 2 n-m -1) -A log(s) = m≥0 N n+m 2 -(n+m) 2 m (s 2 -m -1) -A log(s).
At this point notice that x(a 1/x -1) ≤ a, for all x ≥ 1, a > 0, which together with the hypothesis on

N n gives log P (Y n,k ≥ A) ≤ m≥0 2 -(n+m) s -A log(s) = 2 2 -1 s2 -n -A log(s). Now set s = 2 n 2 , A = 4 in the last inequality to get log P (Y n,k ≥ A) ≤ 2 2 -1 2 -n 2 -2n log(2).
Hence, P (Y n,k ≥ 4 ) ≤ C( )2 -2n . In view of an application of the Borel-Cantelli Lemma we compute n≥0

2 n k=1 P (Y n,k ≥ A) ≤ C( ) n≥0 2 n × 2 -2n < ∞.
Hence, by the Borel-Cantelli Lemma, the event Y n,k ≥ A can happen for at most a finite number of indices (n, k) almost surely. In particular the Carleson measure constant of dµ Λ is almost surely at most A except for a finite number of Carleson windows.

2.2. An example. Here we construct an example showing that we cannot choose β = 1 in the theorem. For γ > 1, let

N n = 2 n n γ , n ≥ 1. Clearly n≥1 2 -n N n < +∞.
For n ∈ N * and k = 0, 1, . . . , 2 n -1 define the event

A N n,k = n+N m=n (X n,k,m ≥ 2 m-n ).
Since X n,k,m are mutually independent for fixed n, k, we have

P (A N n,k ) = n+N m=n P (X n,k,m ≥ 2 m-n ).
The following well known elementary lemma will be very useful (it is essentially approximation of the binomial law by the Poisson law). We refer for instance to [START_REF] Billingsley | Probability and measure[END_REF] for the material on probability theory -essentially elementary -used in this paper.

Lemma 2.1. If X is a binomial random variable with parameters p, N, then for every s = 0, 1, 2, . . .,

lim N →∞ pN →0 P (X = s) (pN ) s = lim N →∞ pN →0 P (X ≥ s) (pN ) s = 1 s! .
Note that in view of the Lemma we can replace in

A N n,k the condition X n,k,m ≥ 2 m-n by X n,k,m = 2 n-m . Since in our situation N = 2 m /m γ → ∞ and pN = N m /2 n ≤ 2 m-n /m γ → 0 (n ≤ m ≤ n + N , N fixed), we get P (A N n,k ) = n+N m=n P (X n,k,m = 2 m-n ) n+N m=n 1 (2 m-n )! N m 2 n 2 m-n = n+N m=n 1 (2 m-n )! 2 m-n m γ 2 m-n = N m=0 1 (2 m )! 2 m (m + n) γ 2 m 1 n 2 N +1 γ . (2.1)
The crucial observation here is that for every fixed N this probability goes polynomially to zero. In particular

n≥0 2 n -1 k=0 P (A N n,k ) ∼ n≥0 2 n n 2 N +1 γ = +∞.
We need to apply a reverse version of the Borel-Cantelli lemma. This works for events which are independent. However this is not the case for A N n,k and A N i,j when the corresponding dyadic annuli meet. Since A N n,k intersects the annuli A l , n ≤ l ≤ n+N and A N i,j intersects A l , i ≤ l ≤ i + N , the events A N n,k and A N i,j are dependent when [n, n + N ] ∩ [i, i + N ] = ∅, i.e. when |n -i| ≤ N + 1. We will appeal to a more general version of the Borel-Cantelli Lemma, see Lemma 6.1. This requires that lim inf

M →+∞ i,j,k,n≤M P (A N i,j ∩ A N n,k ) k,n≤M P (A N n,k ) 2 ≤ 1.
The difference here is of course made by the elements with |n -i| ≤ N + 1 since for |n -i| > N + 1, we have P (A N i,j ∩ A N n,k ) = P (A N i,j ) × P (A N n,k ). In particular i,j,k,n≤M

P (A N i,j ∩ A N n,k ) = k,n≤M P (A N n,k ) 2 + |n-i|≤N +1 i,j,k,n≤M P (A N i,j ∩ A N n,k ) - |n-i|≤N +1 i,j,k,n≤M P (A N i,j ) × P (A N n,k ).
For fixed N we can assume i and n big so that N N n , N i . Note that

P (A N i,j ∩ A N n,k ) = P (A N i,j ) × P (A N n,k |A N i,j
), and we have to estimate

P (A N n,k |A N i,j
). The idea is to observe that the conditional probability essentially behaves like the unconditional one, i.e. the knowledge of A N i,j does not interfer too much on the probability of A N n,k . More precisely, if |n -i| ≤ N + 1, the fact that A N i,j has occured reduces the number of points available for A N n,k (in the annulus A m ) by at most 2 N which we can again assume neglectible with respect to N m , n ≤ m ≤ n + N .

One has to pay a little bit attention here. A priori, it could happen that S i,j ⊂ S n,k or S n,k ⊂ S i,j (in these cases the knowledge of A N i,j has some indicidence to that of A N n,k or vice versa), but this requires |j -k| ≤ 2 N which is uniformly bounded. The corresponding sum of probabilities is thus also uniformly bounded and dividing by the square of eventually divergent partial sums makes these terms neglectible. So we can assume that this pathological situation does not occur.

Then, in order to estimate the probability of A N n,k , under the condition A N i,j , we can run a similar computation as in (2.1) with N m replaced by N m -N = N m (1-N/N m ) (for those m for which the annulus A m lies in both Carleson boxes; for the others we keep N m ), which yields a comparable probability:

P (A N i,j ∩ A N n,k ) = (1 + ε n,k,i,j )P (A N i,j ) × P (A N n,k )
where ε n,k,i,j → 0 when i, j, n, k get big. Then

|n-i|≤N +1 i,j,k,n≤M P (A N i,j ∩ A N n,k ) - |n-i|≤N +1 i,j,k,n≤M P (A N i,j ) × P (A N n,k ) = n,k≤M P (A M n,k )(1 -P (A M n,k ) + 0<|n-i|≤N +1 i,j,k,n≤M ε n,k,i,j P (A N i,j ) × P (A N n,k ), which, when dividing through k,n≤M P (A N n,k ) 2
→ +∞, M → ∞, goes to 0. From Lemma 6.1 we conclude that for every N there exists infinitely many n, k such that µ Λ (S n,k ) ≥ N + 1, which concludes the example.

Interpolation in

D α , 0 < α < 1.
We start this section with the proof of Theorem 1.4 which we recall here for convenience.

Theorem. Let 0 < α < 1. If (3.1) n 2 -(1-α)n N n < ∞ then µ Λ is almost surely a Carleson measure for D α .
Recall also that

µ Λ = λ∈Λ 1 k λ 2 α δ λ ,
where

k λ 2 α =      log 1 1 -|λ| 2 , α = 1, 1 (1 -|λ| 2 ) 1-α , 0 ≤ α < 1.
The case α = 1 will be useful later in the study of the classical Dirichlet space D.

It shall be observed that the proof presented below does not work for the Hardy space case α = 0, for which we have seen that it is possible to construct sequences (r n ) satisfying the Blaschke condition, but the associated sequences Λ(ω) are not almost surely interpolating.

We will need Stegenga's characterization of Carleson measures for Dirichlet spaces which involves capacities. In connection with this result we recall the the following three facts. Once these facts collected, the proof is essentially the same as in the Hardy space.

By Cap α we will mean logarithmic capacity when α = 1 and just α-capacity when α ∈ (0, 1) (we refer the reader to e.g. [26, p.19], [START_REF] El-Fallah | A primer on the Dirichlet space[END_REF] for more information on capacities). The first fact we would like to recall are the following known estimates (see [26, p.19])

Cap α (I)    |I| 1-α , 0 < α < 1, log e |I| -1 , α = 1, (3.2)
where I is an interval.

The second fact is Stegenga's result (see e.g. [26, p.19]). In view of our discussion on D we should emphasize that the above discussion is true for every α ∈ (0, 1].

Consider now the case α ∈ (0, 1). Then (3.5) becomes

λ∈S I (1 -|λ| 2 ) 1-α ≤ c|I| 1-α .
As in the Hardy space, it is enough to discuss this inequality for dyadic intervals I = I n,k , and covering again S n,k by dyadic arcs we obtain the following random variable

Y α n,k = 2 (1-α)n m≥n 2 -(1-α)m X n,m,k .
We now get using again x(a

1/x -1) ≤ a for x ≥ 1, a > 0, log P (Y α n,k ≥ A) = log P (s Y n,k ≥ s A ) ≤ log 1 s A G Y n,k (s) = m≥n N m log(1 -2 -n + 2 -n s 2 (1-α)(n-m) ) -A log(s) ≤ 2 -n m≥n N m (s 2 (1-α)(n-m) -1) -A log(s) ≤ 2 -αn m≥n N m 2 -(1-α)m 2 (1-α)(m-n) (s 2 (1-α)(n-m) -1) ≤s -A log(s). ≤ 2 -αn sc n -A log(s),
where c n = m≥n 2 -(1-α)m N m is the value of the remainder sum which tends to zero, and which we thus can assume less than 1. Now set s = 2 αn , A = 4 α in the last inequality to get log P (Y n,k ≥ A) ≤ 1 -2n log(2).

Hence, P (Y n,k ≥ 4 α ) ≤ C(α)2 -2n , and we conclude as in the Hardy space case. We are now in a position to prove Theorem 1.5.

Proof of Theorem 1.5. (i) Let 0 < α < 1/2.
If Λ is interpolating almost surely, then it is separated almost surely, which implies

n 2 -n N 2 n < +∞. If n 2 -n N 2
n < +∞, then Λ is almost surely separated. Moreover, the condition implies that N n ≤ c2 n/2 for some constant c > 0, and hence

n 2 -(1-α) N n ≤ n 2 -(1/2-α) < +∞.
Theorem 1.4 implies that µ Λ is almost surely a Carleson measure so that Λ is almost surely interpolating.

The remaining cases of a divergent sum and interpolation with zero probability follows from the above and the Kolomogorov 0-1 law.

(ii) Consider the case 1/2 ≤ α < 1. If Λ is interpolating almost surely, then it is a zero sequence almost surely, which implies n 2 -(1-α)n N n < +∞ by Theorem 1.2.

Suppose n 2 -(1-α)n N n < +∞. Again by Theorem 1.4 we that µ Λ is almost surely a Carleson measure.

The condition clearly implies also that n 2 -n N 2 n < +∞, which further yields that the sequence is almost surely separated. Hence it is almost surely an interpolating sequence.

Agian, the remaining cases of a divergent sum and interpolation with zero probability follows from the above and the Kolomogorov 0-1 law.

Separated random sequences for the Dirichlet space

We will now prove the separation result in D.

Proof of Theorem 1.7. Separation with probability 0. Assume that for all γ ∈ (1/2, 1) we have sup k 2 -γk N 2 k = ∞. As it turns out, under the condition of the Theorem, separation already fails in dyadic annuli (without taking into account radial Dirichlet separation).

Assume now that γ l → 1 as l → ∞ and sup k 2 -γ l k N 2 k = ∞ for every l. For each k = 1, 2 . . ., let

I k = [1 -2 -k+1 , 1 -2 -k ). Define Ω (l) k = {ω : ∃(i, j), i = j with ρ i , ρ j ∈ I k and |θ i (ω) -θ j (ω)| ≤ π2 -γ l k }. In view of (1.7), if ω ∈ Ω (l)
k , this means that in the dyadic annulus A k there are at least two points close in the Dirichlet metric. To be more precise, if ω ∈ Ω (l) k , then there is a pair of distinct point λ i (ω) and λ j (ω) such that

|λ i |, |λ j | ∈ I k and | arg λ i (ω) -arg λ j (ω)| ≤ π2 -γ l k . Hence (1 -|λ i | 2 )(1 -|λ j | 2 ) |1 -λ i λ j | 2 ≥ c 2 -2k 2 -2k + π2 -2γ l k ,
where the constant c is an absolute constant. Hence (1

-|λ i | 2 )(1 -|λ j | 2 ) |1 -λ i λ j | 2 ≥ c 1 1 + π2 k(1-γ l ) ≥ c 2 -k(1-γ l ) ≥ c (1 -|λ i |) 1-γ l .
Absorbing c into a suitable change of the power δ 2 l := 1 -γ l into δ l 2 (which can be taken by choosing for instance 2δ l > δ l > δ l provided k is large enough), then by (1.7) ρ D (λ i (ω), λ j (ω)) ≤ δ l . Our aim is thus to show that for every l ∈ N, we can find almost surely λ i (ω) = λ j (ω) such that ρ D (λ i (ω), λ j (ω)) ≤ δ l , i.e. P (Ω (l) k ) = 1. (Note that δ l → 0 when l → +∞.) Let us define a set E := {j : 2 -γ l j-1 N j ≤ 1}. Observe that when k / ∈ E, then at least two points are closer than π2 -γ l k (this is completely deterministic), so that in that case P (Ω (l) k ) = 1. Hence if E N, then we are done.

Consider now the case E = N, and let k ∈ E = N. We will use the Lemma on the probability of an uncrowded road [12, p. 740], which states

P (Ω (l) k ) = 1 -(1 -N k 2 -γ l k-1 ) N k -1 (since E = N this is well defined).
We can assume that N k ≥ 2 (since obviously k:

N k <2 2 -γ l k N k < ∞). In particular N 2 k /2 ≤ N k (N k -1) ≤ N 2 k . Since log(1 -x) ≤ -x, we get k:N k ≥2 (N k -1) log(1 -N k 2 -γ l k-1 ) ≤ - k:M k ≥2 (N k -1)N k 2 -γ l k-1 ≤ - 1 2 k:N k ≥2 N 2 k 2 -γ l k-1 = -∞
by assumption. Hence, taking exponentials in the previous estimate,

k∈E,N k ≥2 (1 -N k 2 -γ l k-1 ) N k -1 = 0,
which implies, by results on convergence on infinite products, that

k P (Ω (l) k ) = ∞.
Since the events Ω (l) k are independent, by the Borel-Cantelli Lemma, P (lim sup Ω (l)

k ) = 1, where lim sup Ω (l) k = n≥1 k≥n Ω (l) k = {ω : ω ∈ Ω (l)
k for infinitely many k}.

In particular, since the probability of being in infinitely many Ω (l) k is one, there is at least one Ω (l) k which happens with probability one. So that again P (Ω

(l) k ) = 1.
As a result, the probability that the sequence is δ l -separated in the Dirichlet metric is zero for every l. Since δ l → 0 when l → +∞, we deduce that P (ω : {λ(ω)} is separated for D) = 0.

Separation with probability 1. Now we assume that k 2 -γk N 2 k < +∞ for some γ ∈ (1/2, 1). Let us begin defining a neighborhood in the Dirichlet metric. For that, fix η > 1 and α ∈ (0, 1). Given λ ∈ Λ, so that for some k, λ ∈ A k . Consider Our aim is to prove that under the condition k 2 -γk N 2 k < +∞, there exists η > 1 and α ∈ (0, 1) such that T η,α λ does not contain any other point of Λ except λ, and this is true for every λ ∈ Λ with probability one. For this we need to estimate

T η,α λ = {z = re it : (1 -|λ|) η ≤ 1 -r ≤ (1 -|λ|) 1/η , |θ -t| ≤ (1 -r) α }.
P (T η,α λ ∩ Λ = {λ}).

Let us cover

T η,α λ = ηk j=k/η (T η,α λ ∩ A j ),
and we need that for every j ∈

[k/η, ηk] \ {k}, (T η,α λ ∩ A j ) ∩ Λ = ∅ and (T η,α λ ∩ A k ) ∩ Λ = {λ}. Note that X j = #(T η,α λ ∩ A j ∩ Λ) ∼ B(N j , 2 -αj ), j = k, and X k ∼ B(N k -1, 2 -αk
) (since we do not count λ in the latter case). Hence, since the arguments of the points are independent, we have

P (T η,α λ ∩ Λ = {λ}) = P ηk j=k/η,j =k (X j = 0) ∩ (X k = 1) = j=ηk j=k/η,j =k P (X j = 0) × P (X k = 1).
From the binomial law we have P (X j = 0) = (1 -2 -αj ) N j , for j ∈ [k/η, ηk] \ {k}. Also, assuming 0 < γ < α < 1, we have N j 2 -αj = o(j), so that

P (X j = 0) = (1 -2 -αj ) N j ∼ 1 -N j 2 -αj . Moreover P (X k = 1) = N k 2 -αk (1 -2 -αk ) N k -1 ∼ N k 2 -αk .
Hence we get

P (T η,α λ ∩ Λ = {λ}) ∼ exp j=ηk j=k/η,j =k ln(P (X j = 0) × N k 2 -αk ∼   1 - j=ηk j=k/η,j =k N j 2 -αj   × N k 2 -αk .
Again we use γ < α < 1 to see now that the sum ηk j=k/η,j =k N j 2 -αj is convergent and goes to zero when k → ∞. This shows in particular that the fact of considering the event of having points in neighboring annuli of A k containing λ can be neglected. Hence

P (T η,α λ ∩ Λ = {λ}) ∼ N k 2 -αk .
We now sum over all λ ∈ Λ by summing over all dyadic annuli A k and the N k points contained in each annuli:

λ∈Λ P (T η,α λ ∩ Λ = {λ}) ∼ k∈N N k × N k 2 -αk = k∈N N 2 k 2 -αk .
For α > γ, this sum converges by assumption. Using the Borel-Cantelli lemma we deduce that T η,α λ ∩ Λ = {λ} for all but finitely many λ with probability one. Obviously these finitely many neighborhoods T η,α λ contain finitely many points between which a lower Dirichlet distance exists. This achieves the proof of the separation.

It should be observed that the above proof only involves α but not η, so that it is the separation in the annuli which dominates the situation.

Proof of Theorem 1.8

In order to prove Theorem 1.8 we will use a method similar to the one for D α . Let us again observe that interpolation implies the zero sequence condition, so that by Bogdan's result, if Λ is almost surely interpolating then n N n /n < +∞.

So suppose now n N n /n < +∞. As mentioned in the proof of Theorem 1.5, it is enough to check that we almost surely have (3.5), now with α = 1. So the condition translates to Setting s = 2 n/2 and A = 4 the above calculation gives

P (Y n,k > 4) C2 -2n .
Again, an application of the Borel-Cantelli Lemma concludes the proof. As already mentioned in preceding cases, the Kolmogorov 0-1 law allows to get also the cases of a divergent sum and interpolation with probability 0. The second Lemma is due to Nagel, Rudin and Shapiro [START_REF] Nagel | Tangential boundary behavior of functions in Dirichlettype spaces[END_REF][START_REF] Nagel | Tangential boundary behavior of harmonic extensions of Lp potentials[END_REF] who discussed tangential approach regions of functions in D α . Proof of Theorem 1.2. In view of our preliminary observations, we are essentially interested in the converse implication. So suppose n 2 -(1-α)n N n = +∞ or equiv-(6.2)

n (1 -ρ n ) 1-α = +∞.
We have to show that Λ is not a zero sequence almost surely. For this, introduce the intervals I = (e -i(1-ρ ) 1-α , e i(1-ρ )

  2 n -1 be dyadic intervals and S n,k = S I n,k the associated Carleson window. In order to check the Carleson measure condition for a positive Borel measure µ on D it is clearly sufficient to check the Carleson measure condition for windows S n,k :µ(S n,k ) ≤ C|I n,k | = C2 -n ,for some fixed C > 0 and every n ∈ N, k = 0, . . . , 2 n -1. Given n, k and m ≥ k let X n,m,k be the number of points of Λ contained in S n,k ∩ A m (we stratify the Carleson window S n,k into a disjoint union of layers S n,k ∩ A m ). Since A m contains N m points and the (normalized) length of I n,k is 2 -n , we have X n,m,k ∼ B(2 -n , N m ) (binomial law). In order to show that dµ Λ is almost surely a Carleson measure we thus have to prove the existence of C such that

Theorem 3 . 1 .

 31 A nonnegative Borel measure µ on D is a Carleson measure for D α , 0 < α ≤ 1, if and only if there exists a constant c > 0 such that n j=1 µ(S I j ) ≤ c Cap α n j=1 I j , (3.3) for each finite collection of disjoint subarcs I 1 , I 2 , . . . , I n of the unit circle, and arbitrary n. The last fact is the following observation. There exists a universal constant c such that for every finite collection I 1 , I 2 , ..., I n of subsarcs of T, and I an arc of T, with |I| = n j=1 |I j |, we have Cap α (I) ≤ c Cap α n j=1 I j . (3.4) Concerning (3.4), we refer to [15, Theorem 2.4.5] for the case α = 1. The general case α ∈ (0, 1) is shown exactly in the same way as for α = 1. Proof of Theorem 1.4. We can now essentially repeat the argument from the Hardy space case. In view of (3.4), in order to apply Theorem 3.1, it is sufficient to show that the condition of Theorem 1.4 implies almost surely n j=1 µ(S I j ) ≤ c Cap α (I), where I is an interval of length n j=1 |I j |. Since the distribution of points in S J for an arc J does not depend on its position on T, we can assume that I j are adjacent intervals. Then, setting I = n j=1 I j we have n j=1 S I j ⊂ S I , and for the application of Stegenga's theorem it is enough to show that almost surely µ(S I ) ≤ c Cap α (I). (3.5)

Figure 1 .

 1 Figure 1. Dirichlet neighborhood.

Figure 1

 1 Figure 1 represents the situation.Our aim is to prove that under the condition k 2 -γk N 2 k < +∞, there exists η > 1 and α ∈ (0, 1) such that T η,α

1 ,

 1 which, using the usual dyadic discretizationI = I n,k , translates to m≥n 1 m X n,m,k ≤ C 1 n almost surely.This leads to estimate the tail of the random variablesY n,k = m≥n n m X n,m,k .To do that, introduce again two positive parameters s, A. Using the formula for the generating function of a binomially distributed random variable and Markov's inequality we can estimate as follows logP (Y n,k ≥ A) = log P (s Y n,k > s A ) ≤ m≥n N m log(1 -2 -n + 2 -n s ( n m ) ) -A log(s) ≤ 2 -n m≥n N m m m n (s ( n m ) -1)n -A log(s) ≤ n2 -n s m≥n N m m -A log(s).

6 .

 6 Annex : Proof of Theorem 1.[START_REF] Agler | Pick interpolation and Hilbert function spaces[END_REF] Carleson proved in[START_REF] Carleson | On a class of meromorphic functions and its associated exceptional sets[END_REF] Theorem 2.2] that, for 0< α < 1, if λ∈Λ (1 -|λ|) 1-α < ∞then the Blaschke product B associated to Λ belongs to D α . So the sufficiency part of Theorem 1.2 follows immediately from this result (and is moreover deterministically true).For the proof of the converse we will need the following two lemmas. The first one is a version of the Borel-Cantelli Lemma[START_REF] Billingsley | Probability and measure[END_REF] Theorem 6.3]. Lemma 6.1. If {A n } is a sequence of measurable subsets in a probability space (X, P ) such that P (A n ) = ∞ and (6.1) lim inf n→∞ j,k≤n P (A j ∩ A k ) [ k≤n P (A k )] 2 ≤ 1, then P (lim sup n→∞ A n ) = 1.

Lemma 6 . 2 .

 62 Let f ∈ D α , 0 < α < 1. Then, for a.e. ζ ∈ T, we have f (z) → f * (ζ) as z → ζ in each region |z -ζ| < κ(1 -|z|) 1-α ,(κ > 1).

2 = lim inf n→∞ 1 +

 21 1-α ) and let F = e iθ I . Denoting by m normalized Lebesgue measure on T, observe thatm(F ) = m(I ) = (1 -ρ ) 1-α . We have for every ϕ ∈ F , λ ∈ Ω κ,ϕ = {z ∈ D : |z -e iϕ | < κ(1-|z|) 1-α }. By Lemma 6.2 it suffices to prove that | lim sup F | > 0 a.s. (the latter condition means that there is a set of strictly positive measure on T to which Λ accumulates in Dirichlet tangential approach regions according to Lemma 6.2, which is of course not possible for a zero sequence). Let E denote the expectation with respect to the Steinhaus sequence (θ n ). By Fubini's theorem we have E[m(F j ∩ F k )] = m(I j )m(I k ), j = k, (the expected size of intersection of two intervals only depends on the product of the length of both intervals). By Fatou's Lemma and (6.2)E lim inf n→∞ j,k≤n m(F j ∩ F k ) [ k≤n m(F k )] 2 ≤ lim inf n→∞ E j,k≤n m(F j ∩ F k ) [ k≤n m(F k )] 2 = lim inf n→∞ j,k≤n E[m(F j ∩ F k )] [ k≤n m(F k )] 2 = lim inf n→∞ j,k≤n,j =k m(I j )m(I k ) + k≤n m(I k ) [ k≤n m(I k )] k≤n m(I k )(1 -m(I k )) [ k≤n m(I k )]2 . Now, since 1 -m(I k ) → 1, and by (6.2), keeping in mind that m(I k ) = (1 -ρ k ) 1-α , we have lim n→∞ k≤n m(I k )(1 -m(I k )) [ k≤n m(I k )] 2 = 0.

This implies that (6.1) holds on a set B of positive probability and hence, by the zeroone law, on a set of probability one. From Lemma 6.1 we conclude P (lim sup n→∞ F n ) = 1 a.s., which is what we had to show.

The research of the first author is supported by the fellowship INDAM-DP-COFUND-2015 "IN-dAM Doctoral Programme in Mathematics and/or Applications cofunded by Marie Sklodowska-Curie Actions" # 713485. The research of the second and third authors are partially supported by the project ANR-18-CE40-0035 and by the Joint French-Russian Research Project PRC CNRS/RFBR 2017-2019. The research of the last author is partially supported by National Science Foundation grants DMS # 1800057 and DMS #1560955.