arXiv subjects

Michael Ren

Publications and source records attributed to Michael Ren.

6 recordsLinked to original sources

arXiv2026

Free energy of Ising models under a spectral condition

A sequence of sparse weighted graphs $(G_N:N\ge 1)$ indexed by the number of vertices $N$ is said to be left-convergent if all (suitably weighted) subgraph counts converge to a limit as $N\to\infty$. This notion generalizes in a natural way Benjamini-Schramm's definition of local weak convergence. A broad research agenda aims at determining which `global' graph properties are determined by left or local weak convergence (in other words, which of these properties are in fact local). We prove that, under a spectral condition on the weighted adjacency matrix ${\boldsymbol A}_N$ of graph $G_N$, the free energy density of the Ising model on this weighted graph is continuous in the left convergence topology (and hence is a local function). Our proof uses the decomposition of the Ising measure as a log-concave combination of product measures, and of the rapid mixing of Langevin dynamics for log-concave measures. As applications, we derive new limit theorems for the free energy density of spin glasses, antiferromagnets, and magnetization constrained ferromagnetic models, on locally tree-like graphs.

math.PR↗

arXiv2023

Stanley-Wilf Limits for Patterns in Rooted Labeled Forests

Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests. We prove a forest analogue of the Stanley-Wilf conjecture for avoiding a single pattern as well as certain other sets of patterns. Our techniques are analytic, easily generalizing to different types of pattern avoidance and allowing for computations of convergent lower bounds of the forest Stanley-Wilf limit in the cases covered by our result. We end with several open questions and directions for future research, including some on the limit distributions of certain statistics of pattern-avoiding forests.

math.CO↗

arXiv2022

Locality of critical percolation on expanding graph sequences

We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probability for the emergence of an infinite component on $G$ coincide with the critical probability for the emergence of a linear-sized component on $G_n$? In this short article we give a positive answer provided the graphs $G_n$ satisfy an expansion condition, and the limiting graph $G$ has finite expected root degree. The main result of Benjamini, Nachmias, and Peres (2011), where this question was first formulated, showed the result assuming the $G_n$ satisfy a uniform degree bound and uniform expansion condition, and converge to a deterministic limit $G$. Later work of Sarkar (2021) extended the result to allow for a random limit $G$, but still required a uniform degree bound and uniform expansion for $G_n$. Our result replaces the degree bound on $G_n$ with the (milder) requirement that $G$ must have finite expected root degree. Our proof is a modification of the previous results, using a pruning procedure and the second moment method to control unbounded degrees.

math.PR↗

arXiv2020

Circular Trace Reconstruction

Trace reconstruction is the problem of learning an unknown string $x$ from independent traces of $x$, where traces are generated by independently deleting each bit of $x$ with some deletion probability $q$. In this paper, we initiate the study of Circular trace reconstruction, where the unknown string $x$ is circular and traces are now rotated by a random cyclic shift. Trace reconstruction is related to many computational biology problems studying DNA, which is a primary motivation for this problem as well, as many types of DNA are known to be circular. Our main results are as follows. First, we prove that we can reconstruct arbitrary circular strings of length $n$ using $\exp\big(\tilde{O}(n^{1/3})\big)$ traces for any constant deletion probability $q$, as long as $n$ is prime or the product of two primes. For $n$ of this form, this nearly matches what was the best known bound of $\exp\big(O(n^{1/3})\big)$ for standard trace reconstruction when this paper was initially released. We note, however, that Chase very recently improved the standard trace reconstruction bound to $\exp\big(\tilde{O}(n^{1/5})\big)$. Next, we prove that we can reconstruct random circular strings with high probability using $n^{O(1)}$ traces for any constant deletion probability $q$. Finally, we prove a lower bound of $\tilde{\Omega}(n^3)$ traces for arbitrary circular strings, which is greater than the best known lower bound of $\tilde{\Omega}(n^{3/2})$ in standard trace reconstruction.

cs.DS↗

arXiv2020

Wilf Equivalences for Patterns in Rooted Labeled Forests

Building off recent work of Garg and Peng, we continue the investigation into classical and consecutive pattern avoidance in rooted forests, resolving some of their conjectures and questions and proving generalizations whenever possible. Through extensions of the forest Simion-Schmidt bijection introduced by Anders and Archer, we demonstrate a new family of forest-Wilf equivalences, completing the classification of forest-Wilf equivalence classes for sets consisting of a pattern of length 3 and a pattern of length at most $5$. We also find a new family of nontrivial c-forest-Wilf equivalences between single patterns using the forest analogue of the Goulden-Jackson cluster method, showing that a $(1-o(1))^n$-fraction of patterns of length $n$ satisfy a nontrivial c-forest-Wilf equivalence and that there are c-forest-Wilf equivalence classes of patterns of length $n$ of exponential size. Additionally, we consider a forest analogue of super-strong-c-Wilf equivalence, introduced for permutations by Dwyer and Elizalde, showing that super-strong-c-forest-Wilf equivalences are trivial by enumerating linear extensions of forest cluster posets.

math.CO↗

arXiv2019

Quasi-Invariants in Characteristic $p$ and Twisted Quasi-Invariants

The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.

math.RT↗