We define a new family of commuting operators $F_k$ in Khovanov–Rozansky link homology, similar to the action of tautological classes in the cohomology of character varieties. We prove that $F_2$ satisfies “hard Lefshetz property” and hence exhibits the symmetry in Khovanov–Rozansky homology conjectured by Dunfield, Gukov, and Rasmussen.

Let ${(G_n)}_{n\ge 1}={((V_n,E_n))}_{n\ge 1}$ be a sequence of finite, connected, vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters ${(p_n)}_{n\ge 1}$ in $[0,1]$ is supercritical with respect to Bernoulli bond percolation ${\mathbb{P}}_p^G$ if there exists $\mathit{\epsilon }>0$ and $N<\mathrm{\infty }$ such that

$${\mathbb{P}}_{(1-\mathit{\epsilon })p_n}^{G_n}(\text{the largest cluster contains at least}\mathit{\epsilon }|V_n|\text{vertices})\ge \mathit{\epsilon }$$

for every $n\ge N$ with $p_n<1$. We prove that if ${(G_n)}_{n\ge 1}$ is sparse, meaning that the degrees are sublinear in the number of vertices, then the supercritical giant cluster is unique with high probability in the sense that if ${(p_n)}_{n\ge 1}$ is supercritical, then

$$\underset{n\to \mathrm{\infty }}{lim}{\mathbb{P}}_{p_n}^{G_n}(\text{the second largest cluster contains at least}c|V_n|\text{vertices})=0$$

for every $c>0$. This result is new even under the stronger hypothesis that ${(G_n)}_{n\ge 1}$ has uniformly bounded vertex degrees, in which case it verifies a conjecture of Benjamini (2001). Previous work of many authors had established the same theorem for complete graphs, tori, hypercubes, and bounded degree expander graphs, each using methods that are highly specific to the examples they treated. We also give a complete solution to the problem of supercritical uniqueness for dense vertex-transitive graphs, establishing a simple, necessary, and sufficient isoperimetric condition for uniqueness to hold.

Using work of Antieau and Bhatt, Morrow, and Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $T{C}^{-}$ of quasi-lci rings with bounded torsion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from ${\mathrm{gr}}^{n}T{C}^{-}(-)$ to ${\mathrm{gr}}^{n}TP(-)$ by $L{\mathrm{\Omega }}_{-∕\mathbb{S}}^{<n}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension d proper over $\mathrm{Spec}(\mathbb{Z})$, and let $n\in \mathbb{Z}$ be an arbitrary integer. Together with Weil-étale cohomology with compact support $R{\mathrm{\Gamma }}_{W,c}(\mathcal{X},\mathbb{Z}(n))$, the complex $L{\mathrm{\Omega }}_{\mathcal{X}∕\mathbb{S}}^{<n}$ is expected to give the zeta value $\pm {\mathit{\zeta }}^{\ast }(\mathcal{X},n)$ on the nose. Following Deninger, we define the zeta function $\mathit{\zeta }({\mathcal{X}}_{\mathbb{R}},s)$ of the $\mathbb{R}$-scheme ${\mathcal{X}}_{\mathbb{R}}$ in terms of zeta-regularized determinants. Our main result is a general special value formula for $\pm {\mathit{\zeta }}^{\ast }({\mathcal{X}}_{\mathbb{R}},n)$ in terms of the Bloch conductor $A{(\mathcal{X})}^{d∕2-n}$ and the determinants of $R\mathrm{\Gamma }({\mathcal{X}}_{\mathbb{R},W},\mathbb{Z}(n))$, $L{\mathrm{\Omega }}_{\mathcal{X}∕\mathbb{S}}^{<n}$, and $L{\mathrm{\Omega }}_{\mathcal{X}∕\mathbb{S}}^{<d-n}$. In particular, the passage from the base ring $\mathbb{Z}$ to the base ${\mathbb{E}}_{\mathrm{\infty }}$-ring $\mathbb{S}$ in derived de Rham cohomology is precisely quantified by the factorials in ${\mathit{\zeta }}^{\ast }({\mathcal{X}}_{\mathbb{R}},n)$.

We prove that the Fröhlich polaron has effective mass at least $\frac{{\mathit{\alpha }}^4}{{(log\mathit{\alpha })}^6}$ when the coupling strength α is large. This nearly matches the quartic growth rate $C_{\ast }{\mathit{\alpha }}^4$ predicted by Landau and Pekar, and complements a recent sharp upper bound proved by Brooks and Seiringer. Our proof works with the path integral formulation of the problem and systematically applies the Gaussian correlation inequality to exploit quasiconcavity of the interaction terms.