Tropical Fan Sampler

1. A ring is one loop of the diagram. Its thick violet arc is the top edge, the largest $u$ in that piece; the same letter marks that edge in the diagram. Going round the rest of the ring you meet beads joined by gray bridges, and every bridge is shorter than the top edge.一个环就是图里的一个圈。粗紫弧是顶边，即这块里最大的 $u$；图里同一个字母标出这条边。沿环的其余部分是由灰色桥连起的珠子，每座桥都比顶边短。
2. A bead is a vertex or a smaller piece. A dot is one vertex (an amber stub means it carries an external leg). A thick light segment is a piece that still has loops; it is drawn by the same rule as its own ring, one column to the right.珠子是一个顶点或一块更小的子图。点是单个顶点（琥珀短线表示带外腿）；粗浅色段是仍含圈的子图，它按同样的规则画成右边一列里自己的环。
3. Beads first. The tables first fix only the sizes: at every level, how many beads and how the loops and pendants are shared out (the dashed rings). Then each bead is strung completely before it is used, innermost first: its threads, then its clasp, and a smaller 1PI diagram is finished. Finished beads are strung on the next thread, and the next clasp closes them. Lengths are drawn in the units of the necklace being made (its clasp = 1) and scaled when it is hung in the next one, so this is the same exact draw, told from the inside out.先串珠子。先由表只定下大小：每一层有几颗珠子、圈和吊坠怎么分（虚线的环）。然后每颗珠子在用之前先完整地穿好，从最里面开始：先穿线，再扣上扣子，一个更小的 1PI 图就做好了。做好的珠子再穿到外一层的线上，由外一层的扣子扣起来。长度都以正在穿的那条项链为单位（它的扣子 = 1），挂到外一层时再整体缩放，所以这是同一个精确的抽样，只是从里往外讲。
4. Each Next draws one edge. First a ring's top edge, whose dashed part is the plan for its chain; then, along the chain, its bridges and inner rings in order. The new edge is pink in the diagram, the tree and the bars. The click after the last edge is Weight: it gives $w$, drops the point on the fan and updates the estimate at once; % labels then show how far the true $U$ agrees with the tropical $U^{tr}$, and coral marks where $w$ loses weight.每按一次“下一步”画一条边。先画一个环的顶边（虚线部分是这条链的计划），再沿着链依次画桥和里面的环。新画的边在图、树和柱状图里都是粉色。最后一条边之后的那一下是“权重”：同时给出 $w$、把点落到扇上并更新估计值；百分比显示真实 $U$ 与热带 $U^{tr}$ 的一致程度，珊瑚色标出权重损失的地方。
5. Each Next draws one edge, largest first. Among the edges still left, edge e is the largest with probability $\frac{J(γ∖e)}{ω(γ∖e)\,J(γ)}$, read off the table $J$ over all $2^E$ edge subsets of this one diagram. The first edge gets $α = 1$; each later one is smaller by a factor $ξ^{1/ω(γ)}$ with $ξ$ uniform on $(0,1)$. Violet means the edge closes a loop of what is left, so its $α$ is a factor of $U^{tr}$; gray means it is a bridge of what is left. The order of all $E$ edges is the Hepp sector.每按一次“下一步”画一条边，从大到小。在剩下的边里，边 e 是最大的概率为 $\frac{J(γ∖e)}{ω(γ∖e)\,J(γ)}$，查自这张图全部 $2^E$ 个边子集上的表 $J$。第一条边取 $α = 1$；之后每条边再乘上一个因子 $ξ^{1/ω(γ)}$（$ξ$ 在 $(0,1)$ 上均匀分布）。紫色表示这条边闭合剩余部分的一个圈，它的 $α$ 是 $U^{tr}$ 的因子；灰色表示它是剩余部分的桥。全部 $E$ 条边的顺序就是 Hepp 扇区。
6. The price is the table. $J$ has $2^E$ entries for every diagram: $2^{20} ≈ 10^6$ at $L = 7$, $2^{29} ≈ 5×10^8$ at $L = 10$, and the number of diagrams grows factorially. That is why this mode stops at $L = 7$, and why the paper moves on to global tropical sampling, which never lists diagrams.代价是这张表。每张图的 $J$ 有 $2^E$ 项：$L = 7$ 时 $2^{20} ≈ 10^6$，$L = 10$ 时 $2^{29} ≈ 5×10^8$，而图的数目按阶乘增长。所以这个模式只到 $L = 7$，论文也因此转向从不列举图的全局热带采样。
7. Each Next is one step of the annealing chain. Raise $β$ a little and bank $Δβ\,\text{log}\,w$, or make one move: regrow a ring and everything inside it from the exact tropical rule (teal if accepted, coral if rejected), or redraw one edge. After $K$ steps of $β$ the run ends and its weight $W$ lands on the fan.每按一次“下一步”是退火链的一步。把 $β$ 升高一点并记下 $Δβ\,\text{log}\,w$，或者做一次移动：按精确的热带规则重长一个环和它里面的一切（被接受为青色，被拒绝为珊瑚色），或者重抽一条边。$β$ 走完 $K$ 步后这一轮结束，它的权重 $W$ 落到扇上。

This is all the sampler draws: a tree of rings and one $u$ per edge. The probability of a draw and the tropical weight $U^{tr} = Π\,(\text{top edge of each ring})$ follow from these alone, so no list of diagrams is ever made. The diagram is glued together from the tree only to compute the true $U$, and with it $w = {(\frac{U^{tr}}{U})}^{3/2}\,Π_e\,u_e^{δ}$.

采样器抽的只有这些：一棵环的树，加上每条边一个 $u$。一个样本的概率和热带权重 $U^{tr} = Π$（每个环的顶边）只由它们决定，所以从不需要图的列表。只有在计算真实的 $U$、从而得到 $w = {(\frac{U^{tr}}{U})}^{3/2}\,Π_e\,u_e^{δ}$ 时，才把树拼成一张具体的图。

This is all diagram tropical draws: an order of the $E$ edges (the Hepp sector) and $E−1$ ratios. In that sector $U^{tr}$ is the product of the violet $α$’s, so $w = {(\frac{U^{tr}}{U})}^{3/2}(Σ_e α_e)^{−ω} ≤ 1$, and $s_G = \frac{Γ(ω)\,J(G)}{|\text{Aut}\,G|}\,\text{mean}(w)$. It is exact and precise for one diagram, but it needs the whole table $J$ of this diagram first, and every diagram must be done separately.

单图热带抽的只有这些：$E$ 条边的一个顺序（Hepp 扇区）和 $E−1$ 个比值。在这个扇区里 $U^{tr}$ 是紫色各 $α$ 的乘积，所以 $w = {(\frac{U^{tr}}{U})}^{3/2}(Σ_e α_e)^{−ω} ≤ 1$，$s_G = \frac{Γ(ω)\,J(G)}{|\text{Aut}\,G|}\,\text{mean}(w)$。对一张图它精确而且精度高，但先要算出这张图的整张表 $J$，而且每张图都得单独做一遍。