arXiv subjects
Publications and source records attributed to Asaf Nachmias.
We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.
We investigate the relationship between the space of ends of a unimodular random hyperbolic manifold and the dynamics of its geodesic flow. We show that having two ends of infinite volume implies recurrence, while having infinitely many such ends implies positive drift and entropy. We also provide a transience criterion which applies to deterministic hyperbolic manifolds. Our method relies on studying Delaunay graphs over point processes and on the analytic notion of capacity.
Let $H_n$ be the row space of a signed adjacency matrix of a $C_4$-free bipartite bi-regular graph in which one part has degree $d(n)\to\infty$ and the other part has degree $k+1$ where $k\geq 1$ is a fixed integer. We show that the local limit as $n\to \infty$ of the determinantal process corresponding to the orthogonal projection on $H_n$ is a variant of a Poisson$(k)$ branching process conditioned to survive. This setup covers a wide class of determinantal processes such as uniform spanning trees, Kalai's determinantal hypertrees, hyperforests in regular polytopal complexes, discrete Grassmanians and incidence matroids, as long as their degree tends to $\infty$.
We prove that for any unweighted graph on n vertices the L1 norm of a unit electric current between the endpoints of a random edge is at most 2 log n. Furthermore, we show that on any weighted graph the spectral norm of the entry-wise absolute value of the symmetric transfer-current matrix is at most 2 log n. This bound is tight up to constants and improves the O(log^2 n) bound from [Schild-Rao-Srivastava, SODA '18]. The initial proofs were generated by OpenAI's ChatGPT 5.5 Pro; the authors have verified and rewritten them to enhance readability and provide additional context.
Let $S_k$ be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of $S_k$ converges to a constant depending only the law of the limiting surface.
We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $λ\in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} λn^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $λ-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs.
A nonnegative function on the vertices of an infinite graph G which vanishes at a distinguished vertex o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We survey basic properties of potentials in recurrent networks. In particular, we show that potentials are Lipschitz with respect to the effective resistance metric, and if the potential is unique, then there is a determinantal formula for the harmonic measures from infinity. We also infer from the von Neumann minimax theorem that there always exists a potential tending to infinity.
Let $S$ be a compact hyperbolic surface of genus $g\geq 2$ and let $I(S) = \frac{1}{\mathrm{Vol}(S)}\int_{S} \frac{1}{\mathrm{Inj}(x)^2 \wedge 1} dx$, where $\mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any $k\in \{1,\ldots, 2g-3\}$, the $k$-th eigenvalue $λ_k$ of the Laplacian satisfies \begin{equation*} λ_k \geq \frac{c k^2}{I(S) g^2} \, , \end{equation*} where $c>0$ is some universal constant. We use this bound to prove the heat kernel estimate \begin{equation*} \frac{1}{\mathrm{Vol}(S)} \int_S \Big| p_t(x,x) -\frac{1}{\mathrm{Vol}(S)} \Big | ~dx \leq C \sqrt{ \frac{I(S)}{t}} \qquad \forall t \geq 1 \, , \end{equation*} where $C<\infty$ is some universal constant. These bounds are optimal in the sense that for every $g\geq 2$ there exists a compact hyperbolic surface of genus $g$ satisfying the reverse inequalities with different constants.
A rooted network consists of a connected, locally finite graph G, equipped with edge conductances and a distinguished vertex o. A nonnegative function on the vertices of G which vanishes at o, has Laplacian 1 at o, and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and Nakai in the settings of Euclidean domains and Riemannian surfaces.
We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.
We study the connected components in critical percolation on the Hamming hypercube $\{0,1\}^m$. We show that their sizes rescaled by $2^{-2m/3}$ converge in distribution, and that, considered as metric measure spaces with the graph distance rescaled by $2^{-m/3}$ and the uniform measure, they converge in distribution with respect to the Gromov-Hausdorff-Prokhorov topology. The two corresponding limits are as in critical Erdős-Rényi graphs.
We show that the union of two or more independent uniform spanning forests (USF) on $\mathbb{Z}^d$ with $d\geq 3$ almost surely forms a connected transient graph. In fact, this also holds when taking the union of a deterministic everywhere percolating set and an independent $ε$-Bernoulli percolation on a single USF sample.
We show that the Brownian continuum random tree is the Gromov-Hausdorff-Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the $d$-dimensional torus $\mathbb{Z}_n^d$ with $d>4$, the hypercube $\{0,1\}^n$, and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.
We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on $n$ vertices with minimal degree linear in $n$ is typically of order $\sqrt{n}$. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.
We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to infinity is the Poisson(1) branching process conditioned to survive forever. An extension to "almost" regular graphs and a quenched version are also given.
We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $Θ(\sqrt{n})$. In particular this result applies to expanders, finite tori $\mathbb{Z}_m^d$ of dimension $d \geq 5$, the hypercube $\{0,1\}^m$, and small perturbations thereof.
Given a finite simple triangulation, we estimate the sizes of circles in its circle packing in terms of Cannon's vertex extremal length. Our estimates provide control over the size of the largest circle in the packing. We use them, combined with results from [12], to prove that in a proper circle packing of the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield, the size of the largest circle goes to zero with high probability.
We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov for isoradial graphs. We observe that by the double circle packing theorem, any finite, simple, 3-connected planar map admits an orthodiagonal representation. Our result improves the work of Skopenkov and Werness by dropping all regularity assumptions required in their work and providing effective bounds. In particular, no bound on the vertex degrees is required. Thus, the result can be applied to models of random planar maps that with high probability admit orthodiagonal representation with mesh size tending to 0. In a companion paper, we show that this can be done for the discrete mating-of-trees random map model of Duplantier, Gwynne, Miller and Sheffield.