Origin of symmetry breaking in the grasshopper model (2024)

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  Figure 1

  The stability of flat half-space in $N$ dimensions to plane-wave perturbations of wavenumber $k$. All values are negative except at $k=0$ (which corresponds to translation of the interface and is not a probability-conserving perturbation) and for $kd$ that are multiples of $2\pi$ in the case $N=2$. These values are zero, meaning that there is a marginal stability to perturbations with wavenumber commensurate with $d$ in two dimensions but not an instability. For larger dimension $N$ of space, the result becomes increasingly insensitive to the commensurability of $k$ and $d$.

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  Figure 2

  The coefficient of stability for the disk to small perturbations of $n$-fold symmetry. A large dot is drawn at the smallest value of $d$ where each curve first becomes positive. The black bar shows the point in $d$ at which the transition to disconnected shapes occurs as measured in [1].

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  Figure 3

  The scaled coefficient $\delta p_n$ for increasing $n$ on the disk as a function of $kd=\sqrt{\pi }nd$, along with the result for the semi-infinite plane ($n=\infty$) from (14). As $n$ is increased, the disk result asymptotically approaches that of the semi-infinite plane. Since the finite-$n$ curves tend to the zeros of the limit curve from above, the marginally stable points of the half-plane are destabilized at finite $n$.

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  Figure 4

  First and second most unstable modes $n$ at a given $d$ for $n\le 18$. The blue and yellow lines show the location of the first and second peak in $\delta p_n\left(d\right)$, respectively. The location of the first peak is extremely similar to (3.4) of [1] but not identical. If the cutoff in $n$ is increased, more bands appear at higher $n$. Black markers denote the corresponding numbers of cogs for the optimal solutions found through numerical simulation in Ref.[1] (the same data are shown in the left panel of Fig.5 in Ref.[1]). The numerical results from [1] are in very close agreement with the current prediction.

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  Figure 5

  (Left) Grasshopper probability for lawns with fixed cog number for two values of $d$, obtained numerically for discrete lawns with $10000$ cells. The maxima corresponding to the two leading unstable modes are clearly visible ($n=9$ and $n=17$ for $d=0.4$; $n=7$ and $n=15$ for $d=0.46$) and are marked with thin vertical lines. These are indeed (local) maxima of the grasshopper problem. Horizontal-dashed lines denote the corresponding disk probabilities given by Eq.(3). (Right) The same probability rescaled by the squared amplitude $\epsilon$ of the cogs (symbols with error bars). Solid lines show $\delta p_n\left(d\right)$ from (25), which corresponds roughly with the finite-$\epsilon$ numerical data.

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  Figure 6

  Study of discretization effects. (Left) The exact continuous probability functional $p_{\mu }\left(d\right)$ for the solid 3-ball of unit volume given by Eq.(4) (red solid line) compared with the corresponding discrete $P_{\left\{s\right\}}\left(d\right)$ (black dots) as function of the grasshopper jump distance $d$. For $d\le R_{0,3}$ the 3-ball configuration is the optimal lawn shape. (Right) Relative deviation of $P_{\left\{s\right\}}\left(d\right)$ for the solid 3-ball configuration from the corresponding $p_{\mu }\left(d\right)$ as function of the lattice spacing $h$ for two representative values of the grasshopper jump: $d=0.2\approx 0.3R_{0,3}$ (blue dots and line) and $d=0.5\approx 0.8R_{0,3}$ (red dots and line). For the highest resolutions considered ($M\approx 113000$) the discretization error is well below $0.3\%$. Lines are to guide the eye.

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  Figure 7

  Study of discretization effects. (Left) The exact continuous probability functional $p_{\mu }\left(d\right)$ for the 3-shell where the inner radius is selected as $R_{i,3}=d-R_{0,3}$ (solid blue line) compared with the corresponding discrete $P_{\left\{s\right\}}\left(d\right)$ (black dots) as function of the grasshopper jump distance $d\ge R_{0,3}$. The corresponding probability for the solid 3-ball (red line) is also shown for comparison. For $d>R_{0,3}$ the 3-shell has a higher success probability than the 3-ball. (Right) Relative deviation of $P_{\left\{s\right\}}\left(d\right)$ for the 3-shell configuration from the corresponding $p_{\mu }\left(d\right)$ as function of the lattice spacing $h$ for two representative values of the grasshopper jump: $d=0.94\approx 1.5R_{0,3}$ (blue dots and line) and $d=0.77\approx 1.25R_{0,3}$ (red dots and line). The inner radius for each $d$ is $R_{i,3}=d-R_{0,3}$, as before. For the highest resolutions considered ($M\approx 162000$) the discretization error is below $1\%$. Lines are to guide the eye.

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  Figure 8

  (Left) Cross sectionof the optimal configuration for $d=0.64R_{0,3}$ found numerically for a system with $M=160000$ spins. The configuration has the shape of a solid 3-ball. This configuration was found to be optimal for all $d\le R_{0,3}$. (Right) Cross sectionof the optimal configuration for $d=1.32R_{0,3}$ found numerically for a system with $M=160,000$ spins. If the jump length exceeds $R_{0,3}$ the configurations remain isotropic (for $d\lesssim 1.4R_{0,3}$) but develop a spherical hole in the center; the radius of the hole grows with increasing $d$. Note that the outer radius of the configuration is slightly larger than $R_{0,3}$ to ensure that it has unit volume.

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  Figure 9

  (Left) The optimal inner radius $R_{i,3}$ of the 3-shell vs grasshopper jump. The numerically found inner radii (black dots) match very well the analytical result (red solid line). The value $d-R_{0,3}$ (blue-dashed lines) is shown for comparison. (Right) The corresponding optimal grasshopper probabilities for the optimal inner radius (red-solid line for analytical value and black dots for the numerical value) and for $d-R_{0,3}$ (blue-dashed lines). The isotropic 3-shell ceases to be optimal for jumps exceeding a critical value of approximately $1.4R_{0,3}$.

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  Figure 10

  Histograms of the radial coordinates of the configuration boundary points for, from left to right, $d=1.39R_{0,3}$ (isotropic 3-shell), $d=1.42R_{0,3}$ (eightfold perturbation), and $d=1.58R_{0,3}$ (sixfold perturbation). Results shown were obtained for systems with $M=160000$ spins. The vertical-red lines denote the theoretical values for the optimal (for the respective value of $d$) inner and outer radii of the corresponding 3-shell configurations. The shaded regions mark the $\pm h$ interval around the optimal radii.

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  Figure 11

  Maximal values found numerically for the discrete grasshopper lawn probability $P_{\left\{s\right\}}\left(d\right)$. Results shown were obtained for systems with $M=40000$ spins. Vertical lines denote boundaries between the different regimes. From left to right these are: isotropic solid ball, isotropic shells, eightfold shell perturbations, sixfold shell perturbations, ring with caps, nested crescents. Insets show examples of representative configurations. 3d animations displaying in full the features summarized here are given within the Supplemental Material [14].

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  Figure 12

  The points give the difference in probability between the perturbed disk and the disk as a function of perturbation size $\epsilon$ for $n=2$ and $d=0.98>d_0$. They were computed using numeric integration on the full expression. The solid line gives $\delta p_2\left(d\right){\epsilon }^2$. The results agree at small $\epsilon$, as expected.

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