gpt 6 astra ultra attempt on Math-426

Realizable Limit Shapes in Two-Dimensional First-Passage Percolation

Definitions and mathematical statement

On nearest-neighbor edges of ${\mathbb{Z}}^2$, assign independent weights $t_e\ge0$ with law $\nu$. Passage time is $T(x,y)=\inf_{\gamma:x\to y}\sum_{e\in\gamma}t_e$. Assume $\nu(\{0\})<p_c(2)$ and ${\mathbb{E}}[\min(t_1,t_2,t_3,t_4)^2]<\infty$ for independent copies. The deterministic time-constant norm is obtained by the directional large-distance limit $\mu_\nu(x)=\lim_{n\to\infty}T(0,\lfloor nx\rfloor)/n$, extended continuously and homogeneously, and its unit ball is \[ B_\nu=\{x\in{\mathbb{R}}^2:\mu_\nu(x)\le1\}. \] These compact convex bodies describe the rescaled infected sets in the shape theorem. Characterize exactly which bodies occur as $\nu$ ranges over this class. Lattice reflection and coordinate-permutation symmetries are necessary, but are not asserted here to be sufficient. The minimum-moment condition must not be replaced by a stronger individual second-moment hypothesis.

Formulation and concept references: [S1].

Short English statement

Which deterministic planar shapes can arise from independent nearest-neighbor first-passage times satisfying the specified positivity and minimum-moment assumptions?

Research attempt

Process and chosen route.

This attempt was generated by GPT-6 Astra Ultra on 27 September 2026. We derive necessary geometric constraints, compute a realizable family exactly, and prove a local integrability estimate under the precise minimum-of-four-weights moment hypothesis. The last calculation is intended to prevent a hidden strengthening of that hypothesis during a shape argument. We do not characterize all realizable shapes.

The source [S1] distinguishes independent identically distributed passage times from stationary dependent environments. Its discussion of the inverse shape problem and of the stationary construction makes this distinction explicit. We retain independence and a common edge law throughout. The existence of shapes in a broader dependent model cannot be used as an iid realization theorem.

Symmetry, convexity, and two elementary enclosing bodies.

Under the stated hypotheses the time constant is a norm, as in the formulation and the shape theorem in [S1]. Reflection of either coordinate and interchange of the coordinates preserve the distribution of the iid environment. Applying these symmetries to passage times and then their deterministic limits gives \[ \mu(x_1,x_2)=\mu(\pm x_1,\pm x_2)=\mu(x_2,x_1). \tag{422.1} \] Consequently its unit ball $B$ is compact, convex, has nonempty interior, and is invariant under the eight symmetries of the square. This does not assert invariance under all rotations.

Let $c=\mu(e_1)=\mu(e_2)>0$ and $r=1/c$. Convexity gives \[ \{x:|x_1|+|x_2|\le r\}\subseteq B\subseteq[-r,r]^2. \tag{422.2} \] For the first inclusion, $B$ contains the four points $\pm re_1,\pm re_2$ and hence their convex hull. For the second, if $(a,b)\in B$, reflection gives $(a,-b)\in B$, so their midpoint $(a,0)$ belongs to $B$. Homogeneity of the norm implies $|a|\le r$; the same argument applies to $b$. In particular \[ c\le\mu(e_1+e_2)\le2c. \tag{422.3} \] The upper bound is the triangle inequality. The lower bound follows by writing $e_1$ as the average of $e_1+e_2$ and $e_1-e_2$, whose norms agree by reflection.

These are necessary constraints on every law in the question. They are not a converse construction. In particular, specifying a convex body with these symmetries does not specify an independent edge distribution realizing it.

An exact family and elementary comparisons.

If every edge has deterministic weight $a>0$, every path from $0$ to $z\in\mathbb Z^2$ has at least $\|z\|_1$ edges, and a coordinate path achieves that length. Therefore \[ T(0,z)=a\|z\|_1,\qquad \mu(x)=a\|x\|_1,\qquad B=\{x:\|x\|_1\le1/a\}. \tag{422.4} \] This law satisfies all the hypotheses, so diamonds of every positive scale are realizable. More generally, multiplying every weight by $a$ multiplies every passage time and the norm by $a$, and scales the shape by $a^{-1}$.

If two edge laws admit a coupling with $t_e\le t'_e$ for every edge, then every fixed path has no greater cost in the first environment. Taking infima gives $T\le T'$, and passage to time constants gives $\mu\le\mu'$ and $B\supseteq B'$. A common-uniform quantile coupling provides such an iid coupling whenever the first law is stochastically dominated by the second. Independence is preserved because the uniforms assigned to different edges are independent.

For example, weights in $[a,b]$ with $0<a\le b<\infty$ satisfy \[ a\|x\|_1\le\mu(x)\le b\|x\|_1. \tag{422.5} \] The lower bound holds for every path, and a shortest coordinate path gives the upper bound. These inclusions estimate a shape but usually do not determine it.

Respecting the actual moment assumption.

The condition $\mathbb E[\min(t_1,t_2,t_3,t_4)^2]<\infty$ need not imply even $\mathbb E t_e<\infty$. Thus an argument that bounds $T(0,ne_1)$ by the straight path and then invokes a finite-mean law of large numbers may silently assume more than the problem permits.

There is an explicit local repair. Between $0$ and $e_1$, consider four edge-disjoint paths: the direct edge; the three-edge route through $(0,1),(1,1)$; the three-edge route through $(0,-1),(1,-1)$; and the nine-edge route \[ (0,0),(-1,0),(-1,1),(-1,2),(0,2),(1,2), (2,2),(2,1),(2,0),(1,0). \] Let $S_i$ be their costs and $D=\min_i S_i$. Their edge sets are pairwise disjoint, so the four costs are independent, even though their paths share endpoints. If a path of $L$ edges has cost greater than $t$, at least one edge has weight greater than $t/L$. With $\overline F(u)=\mathbb P(t_e>u)$ and path lengths $1,3,3,9$, \[ \mathbb P(D>t)=\prod_i\mathbb P(S_i>t) \le81\,\overline F(t/9)^4. \tag{422.6} \] For $Y=\min(t_1,t_2,t_3,t_4)$, independence gives $\mathbb P(Y>u)=\overline F(u)^4$. Integrating tails and substituting $t=9u$ yields \[ \mathbb E D^2\le6561\,\mathbb E Y^2<\infty. \tag{422.7} \] In particular the cost of reaching a neighbor has a finite second moment, since $T(0,e_1)\le D$. No moment of an individual edge was substituted for the stated one.

From local detours to a finite directional bound.

Translate the four-path construction by $ke_1$ and denote its minimum cost by $D_k$. The sequence is stationary and depends only on weights in translated copies of a fixed finite rectangle. Variables whose indices are sufficiently far apart use disjoint edge sets and are independent. Splitting the indices into finitely many arithmetic progressions and applying the ordinary iid strong law on each gives \[ \frac1n\sum_{k=0}^{n-1}D_k\longrightarrow\mathbb E D \quad\hbox{almost surely}. \] Concatenating the selected local routes is a walk from $0$ to $ne_1$, so nonnegativity of the weights and the definition of passage time give \[ T(0,ne_1)\le\sum_{k=0}^{n-1}D_k, \qquad \mu(e_1)\le\mathbb E D\le81\sqrt{\mathbb E Y^2}. \tag{422.8} \] Loops in the concatenated walk can be removed without increasing its cost. This proves an explicit finite bound under exactly the original moment hypothesis. It provides no exact value of $\mu(e_1)$ because the true optimum can use other routes on arbitrarily large scales.

For a concrete heavy-tailed example, take $t_e\ge1$ with $\mathbb P(t_e>u)=u^{-3/4}$ for $u\ge1$. Then $\mathbb E t_e=\infty$, while $\mathbb P(Y>u)=u^{-3}$ and \[ \mathbb E Y^2=1+\int_1^\infty2u\,u^{-3}\,du=3. \] Its zero atom is absent, and all hypotheses are satisfied. Equations $422.5$'s lower-bound argument and $422.8$ place its norm between $\|x\|_1$ and $81\sqrt3\,\|x\|_1$. The example demonstrates why the minimum-moment formulation has real content.

A proposed interpolation shortcut fails at the path level.

Multiplying weights has the exact behavior already proved, but adding a constant does not simply add that constant times the displacement norm to passage time. On a square, give the direct edge from $0$ to $e_1$ cost $10$ and a three-edge detour cost zero on each edge. The original passage time is zero. After adding $1$ to every edge, the detour costs $3$, while the direct edge costs $11$; every alternative nearest-neighbor route has at least three edges, so the new passage time is exactly $3$. It is not $0+\|e_1\|_1=1$.

This finite pattern occurs with positive probability in an iid two-point law with a sufficiently small positive atom at zero. It invalidates the alleged pathwise identity; it does not compute the limiting shape of that law. An inverse-shape construction based on linearly interpolating norms by adding edge constants would need a different justification.

Verification and outcome.

The exact checks verify disjointness and lengths of the four routes, the constants in $422.6$--$422.8$, and the finite interpolation counterexample. The symmetry and convexity statements are proved from the norm and iid invariance, while the existence of that norm is the established shape-theorem input in [S1].

UNRESOLVED. We obtain necessary geometry, an exact diamond family, monotone comparisons, and moment-valid local bounds. No sufficiency theorem constructs every admissible-looking body from an iid edge law, and no complete characterization of realizable planar shapes is established.

Sources

• S1 Auffinger, Damron and Hanson, 50 years of first-passage percolation. https://sites.math.duke.edu/~rtd/FPP/50yrsFPP.pdf. Formulation source.