Isosurface-Crossing Probability in Random Fields

• Isosurface-crossing probability is a measure of threshold events in random fields, characterizing connectivity in level-set percolation and local intersection rates.
• The concept is formulated via discrete and metric-graph approaches in the Gaussian Free Field, with distinct behaviors under zero and alternating boundary conditions.
• Analysis using exploration martingales, harmonic measure, and entropic repulsion provides practical insights into scaling laws and anisotropic crossing statistics.

Searching arXiv for the cited papers and closely related topic coverage. arXiv search query: (Ding et al., 2020) Crossing estimates from metric graph and discrete GFF Isosurface-crossing probability denotes a family of probabilistic quantities attached to level sets of random fields and interfaces. In the literature considered here, it appears in three closely related forms: the probability that a level-$0$ set of the two-dimensional Gaussian free field (GFF) contains a horizontal crossing of a rectangle; the probability that a discrete interface remains entirely above the height-$0$ wall in a large region, equivalently avoiding any crossing of that wall; and the Rice-type rate at which a line intersects the isosurface $\{H=\alpha\}$ of an anisotropic stochastic surface (Ding et al., 2020, Caputo et al., 2014, Nezhadhaghighi et al., 2015). These formulations share a common concern with how random geometry interacts with a prescribed threshold, but they probe distinct objects: connectivity of excursion sets, global positivity events, and local line–isosurface intersection densities.

1. Formalizations of the quantity

A first formalization arises in level-set percolation for the GFF. In the rectangle

$$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$

one studies the event that the level set $\{\phi_\delta\ge 0\}$ or $\{\tilde\phi\ge 0\}$ contains a path connecting the left and right sides. The discrete setting uses the lattice $V_\delta=R_L\cap \delta\mathbb{Z}^2$, whereas the metric-graph setting replaces each nearest-neighbor edge by a cable interval, producing the cable system $\widetilde V_\delta$ (Ding et al., 2020).

A second formalization is local rather than global. For a stochastic surface $H(x,y)$, an up-crossing of level $\alpha$ along a direction $0$0 is a line–isosurface intersection where the field crosses $0$1 with positive directional derivative. The corresponding expected count per unit length is denoted $0$2, and its inverse $0$3 is the statistical characteristic spacing between up-crossings at that level (Nezhadhaghighi et al., 2015).

A third formalization is dual to crossing. In the $0$4-dimensional discrete SOS model, the event

$0$5

is the event that the interface does not cross the wall $0$6 anywhere in the region. Its complement is the existence of at least one negative site, hence a crossing of the wall in the large-box sense used in the SOS literature (Caputo et al., 2014).

These uses are not interchangeable. The GFF crossing event is a connectivity probability, the Rice quantity is a crossing rate per unit length, and the SOS positivity event is a global avoidance probability. Their common structure is threshold geometry.

2. GFF level-set crossing on discrete and metric graphs

For the discrete GFF on a finite graph $0$7 with boundary condition $0$8, the mean is the harmonic extension of $0$9 and the covariance is the simple-random-walk Green’s function $\{H=\alpha\}$0. With Dirichlet boundary values fixed, the interior density is

$\{H=\alpha\}$1

For the metric graph GFF $\{H=\alpha\}$2 on $\{H=\alpha\}$3, the field is continuous on cables, its restriction to vertices equals the discrete field, and on each edge, conditioned on the endpoint values, it is a Brownian bridge. The associated energy is

$\{H=\alpha\}$4

This continuous interpolation is central to the comparison of crossing events (Ding et al., 2020).

At level $\{H=\alpha\}$5, the discrete horizontal crossing event is the existence of a nearest-neighbor path in $\{H=\alpha\}$6 connecting the left and right boundary arcs on which $\{H=\alpha\}$7 throughout. The metric-graph version requires a continuous cable path connecting the corresponding arcs with $\{H=\alpha\}$8 everywhere. Under zero boundary conditions, one uses one-step-inward arcs $\{H=\alpha\}$9 and $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$0 to avoid trivial boundary paths. Under alternating boundary conditions, one prescribes

$$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$1

The principal comparison result is that the discrete crossing probability is strictly larger than the metric-graph crossing probability for small mesh. Under zero boundary conditions, there exist $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$2 and $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$3 such that, for all $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$4,

$$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$5

while there exists $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$6 such that

$$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$7

Thus, as $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$8, the discrete crossing probability stays bounded away from $$R_L=\{z\in\mathbb{C}:0<\operatorname{Re}(z)<L,\ 0<\operatorname{Im}(z)<1\},$$9 and $\{\phi_\delta\ge 0\}$0, whereas the metric-graph crossing probability tends to $\{\phi_\delta\ge 0\}$1 at least at rate $\{\phi_\delta\ge 0\}$2 (Ding et al., 2020).

Under alternating $\{\phi_\delta\ge 0\}$3 boundary conditions, both models have crossing probabilities bounded away from $\{\phi_\delta\ge 0\}$4 and $\{\phi_\delta\ge 0\}$5, but the strict comparison persists: there exists $\{\phi_\delta\ge 0\}$6 such that, for sufficiently small $\{\phi_\delta\ge 0\}$7,

$\{\phi_\delta\ge 0\}$8

3. Conformal limits and pivotal obstructions

In the discrete model with alternating boundary conditions at a distinguished value $\{\phi_\delta\ge 0\}$9, the level line converges to $\{\tilde\phi\ge 0\}$0. Consequently,

$\{\tilde\phi\ge 0\}$1

for any conformal map $\{\tilde\phi\ge 0\}$2 with $\{\tilde\phi\ge 0\}$3. The limiting crossing probability is therefore a conformally invariant cross-ratio-type expression, and for rectangles it depends on the aspect ratio through the conformal map (Ding et al., 2020).

The metric-graph model exhibits a distinct obstruction mechanism. Define $\{\tilde\phi\ge 0\}$4 if $\{\tilde\phi\ge 0\}$5 for all $\{\tilde\phi\ge 0\}$6, and $\{\tilde\phi\ge 0\}$7 otherwise. An edge is pivotal if forcing it open creates a crossing while forcing it closed destroys that crossing; a closed pivotal edge is pivotal with $\{\tilde\phi\ge 0\}$8. Under alternating boundary conditions, there exist $\{\tilde\phi\ge 0\}$9 and $V_\delta=R_L\cap \delta\mathbb{Z}^2$0 such that

$V_\delta=R_L\cap \delta\mathbb{Z}^2$1

for all $V_\delta=R_L\cap \delta\mathbb{Z}^2$2 (Ding et al., 2020).

This distinction clarifies a common misconception. The discrete and metric-graph fields agree on vertex values, but they do not induce the same sign-percolation geometry. The metric graph inserts Brownian bridges along edges, creating additional opportunities for the sign to flip inside an edge and thereby generating closed pivotal edges that have no discrete analogue at the same resolution. This suggests that continuity along cables is not a benign interpolation; it changes the threshold geometry in a quantitatively detectable way.

4. Martingale methods, harmonic measure, and entropic repulsion

The comparison between discrete and metric-graph crossings is obtained through exploration martingales for GFF sign clusters. One considers optional sets $V_\delta=R_L\cap \delta\mathbb{Z}^2$3 generated by exploring the excursion set above $V_\delta=R_L\cap \delta\mathbb{Z}^2$4 from a seed boundary arc, and an observable

$V_\delta=R_L\cap \delta\mathbb{Z}^2$5

for a suitable finite set $V_\delta=R_L\cap \delta\mathbb{Z}^2$6 near the target boundary. The Doob martingale is

$V_\delta=R_L\cap \delta\mathbb{Z}^2$7

By the strong Markov property of the metric-graph GFF at optional sets, the field on the unexplored region remains a GFF with boundary determined by the explored set, permitting explicit control of the quadratic variation (Ding et al., 2020).

The key identity is

$V_\delta=R_L\cap \delta\mathbb{Z}^2$8

where $V_\delta=R_L\cap \delta\mathbb{Z}^2$9 is the Green’s function on the initial domain and $\widetilde V_\delta$0 is harmonic measure at time $\widetilde V_\delta$1. On the event of a positive crossing, one constructs an explored boundary path $\widetilde V_\delta$2 such that

$\widetilde V_\delta$3

hence $\widetilde V_\delta$4. A Brownian time-change then turns quadratic-variation growth into quantitative hitting estimates for one-dimensional Brownian motion, which yields the upper bound on metric-graph crossing probabilities.

In the discrete model, a different mechanism dominates. When the exploration does not connect to the target side, the outer boundary of the explored set has strictly negative field with high probability. The paper formulates this as an entropic repulsion estimate: for small $\widetilde V_\delta$5, conditioning on the explored set being negative forces a frontier average $\widetilde V_\delta$6 to be significantly negative with high probability. Combined with the bound

$\widetilde V_\delta$7

this yields that the discrete crossing probability remains bounded away from both $\widetilde V_\delta$8 and $\widetilde V_\delta$9 under zero boundary conditions (Ding et al., 2020).

The contrast is structural. In the metric graph, Brownian-bridge fluctuations inside edges create extra sign-change obstructions; in the discrete graph, entropic repulsion pushes the unexplored frontier away from zero and stabilizes sign clusters.

5. Rice-type crossing statistics for anisotropic stochastic surfaces

For a height field $H(x,y)$0 on a two-dimensional lattice, the crossing-statistics approach studies intersections of the level set $H(x,y)$1 with one-dimensional line scans. Along a direction $H(x,y)$2 at rotation angle $H(x,y)$3, an up-crossing is a crossing with positive slope, and the expected up-crossing count per unit length is $H(x,y)$4. The generalized up-crossing, or generalized roughness, is

$H(x,y)$5

At $H(x,y)$6, $H(x,y)$7 is the total number of up-crossings across all levels along direction $H(x,y)$8 (Nezhadhaghighi et al., 2015).

The basic Rice representation is

$H(x,y)$9

For an isotropic Gaussian field in $\alpha$0 dimensions with mean zero and spectral parameters

$\alpha$1

the paper gives

$\alpha$2

For a one-dimensional line scan through a two-dimensional surface,

$\alpha$3

The framework is explicitly directional. For anisotropic correlated Gaussian surfaces with distinct correlation lengths $\alpha$4 and $\alpha$5, the paper derives

$\alpha$6

Thus the up-crossing density encodes the inverse correlation-length scale along the scan direction. More generally, the analytical relations for anisotropic scaling surfaces involve both directional correlation lengths and directional scaling exponents (Nezhadhaghighi et al., 2015).

Operationally, anisotropy is inferred by rotating the scan directions, computing $\alpha$7 and $\alpha$8 across thresholds, and aggregating them through $\alpha$9 and $0$00. The paper defines a rotation-search statistic $0$01, applies a two-sample $0$02-test with unequal variances, combines $0$03-values across $0$04 into $0$05, and declares anisotropy detected at $0$06 when $0$07. In the ion-beam sputtering example, $0$08 peaks near $0$09, and the corresponding $0$10 indicates significant anisotropy (Nezhadhaghighi et al., 2015).

A useful conceptual distinction follows. Here, “isosurface-crossing probability” is not a connectivity event but an expected line–isosurface intersection rate. This suggests that the same threshold geometry can be probed either globally, through percolative crossing events, or locally, through Rice-type intersection statistics.

6. Wall-avoidance asymptotics, contrasts, and open directions

In the low-temperature $0$11-dimensional discrete SOS model, the relevant threshold event is global positivity above the wall. For the square

$0$12

there exists $0$13 such that for all $0$14,

$0$15

and the same limit holds under the infinite-volume measure $0$16. The same leading asymptotics also hold if $0$17 is replaced by $0$18 for any fixed $0$19 (Caputo et al., 2014).

Writing the box side as $0$20, this becomes

$0$21

up to lower-order corrections. Equivalently,

$0$22

The mechanism is entropic repulsion: conditioned on positivity, the interface typically lifts to height

$0$23

producing $0$24 nested macroscopic level lines surrounding the box. Each level line costs essentially perimeter times $0$25, and an FKG-based monotonicity theorem for staircase ensembles shows that interactions between contours do not reduce the leading free-energy cost (Caputo et al., 2014).

This behavior is qualitatively different from massless continuous-height gradient models. The same source notes that for the two-dimensional GFF, positivity in a shrunken inner square has a $0$26 large-deviation scale, while positivity in the full square is expected to have linear scale $0$27. A plausible implication is that “isosurface-crossing probability” is highly model-dependent even when the threshold is identical: discrete SOS interfaces are controlled by step free energy and macroscopic contour stacks, whereas GFF level sets are controlled by harmonic structure, SLE-type interfaces, and metric-graph sign flips.

Several limitations and open problems remain explicit in the sources. For the GFF comparison, the results are two-dimensional, stated for the square lattice $0$28, and proved for level $0$29 under zero Dirichlet or alternating $0$30 boundary conditions. The explicit crossing limit is available only for the discrete model at $0$31. The zero-boundary metric-graph crossing probability is conjectured to decay as $0$32, but the proved upper bound is only $0$33; the discrete zero-boundary crossing probability is expected to converge to an $0$34 crossing event, but no explicit formula is available; and the role of pivotal vertices in the discrete GFF remains unclear (Ding et al., 2020). For crossing statistics on rough surfaces, the reported framework is applicable beyond the Gaussian case through perturbative corrections, but finite-size effects, resolution limits, and departures from Gaussianity constrain inference from empirical data (Nezhadhaghighi et al., 2015).