arXiv subjects

Shengjing Xu

Publications and source records attributed to Shengjing Xu.

5 recordsLinked to original sources

arXiv2026

Support of Dyson Brownian Motion

We consider beta-Dyson Brownian motion, with $\beta >= 1$, started from a deterministic configuration with uniformly bounded support. Let $\mu_t$ be the semicircular free-convolution flow issued from the initial empirical measure, and set $S_t = supp(\mu_t)$. For every fixed $T, \epsilon > 0$, with probability at least $1 - C \exp(-(log n)^2)$, every particle remains within an epsilon-neighborhood of $S_t$ for all $0 <= t <= T$. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an $o((n \eta)^(-1))$ comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.

math.PR↗

arXiv2026

Counterexamples to Sato's Weak F-Equivalence Conjecture and a Gorenstein Refinement

We disprove Sato's weak $F$-equivalence conjecture for nonsingular projective toric weak Fano varieties in every dimension $d \geq 3$. Our counterexamples are smooth projective crepant models of centered reflexive simplices. The key input is a rigidity property of ray polytopes: if $X_\Sigma$ is nonsingular and complete and $-K_{X_\Sigma}$ is nef, then every nonzero lattice point of $P_\Sigma=\operatorname{Conv}(G(\Sigma))$ is a primitive ray generator. For our models, this rules out every weak-Fano-preserving equivariant blow-up and blow-down throughout the flop class. We then introduce Gorenstein weak $F$-equivalence, generated by projective toric birational zigzags through normal projective Gorenstein toric weak Fano varieties, and formulate a corresponding refinement of Sato's conjecture. We prove this refined conjecture in dimensions $d \leq 3$, as well as for the family of counterexamples constructed above in every dimension. Finally, we show that the refined conjecture implies the inclusion-connectivity of reflexive $d$-polytopes modulo unimodular equivalence, which is known for $d \leq 4$ and remains open for $d \geq 5$. The results were developed with the assistance of GPT-5.6 Sol.

math.AG↗

arXiv2026

Ward identities: a geometric point of view and applications

The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.

math-ph↗

arXiv2026

Lagrangian correspondences for moduli spaces of Higgs bundles and holomorphic connections

On a compact connected Riemann surface $C$ of genus at least $2$, we construct Lagrangian correspondences between moduli spaces of rank-$n$ Higgs bundles (respectively, holomorphic connections) and the Hilbert schemes of points on $T^\ast C$ (respectively, the twisted cotangent bundles of $C$). Central to these constructions are Higgs bundles (respectively, holomorphic connections) which are transversal to line subbundles of the underlying bundles: these naturally induce divisors on $C$ together with auxiliary parameters, namely lifts to divisors on spectral curves for Higgs bundles and residue parameters of apparent singularities for holomorphic connections. We discuss the evidence showing that the Dolbeault geometric Langlands correspondence is generically realized by these Lagrangian correspondences; we expect that the de Rham geometric Langlands correspondence can be realized by their quantization, following Drinfeld's construction of Hecke eigensheaves. We also discuss the relations of our constructions to various topics, including reductions of Kapustin-Witten equations, the conformal limit, separation of variables, and degenerate fields in conformal field theories.

math.AG↗

arXiv2025

Annulus crossing formulae for critical planar percolation

We derive exact formulae for three basic annulus crossing events for the critical planar Bernoulli percolation in the continuum limit. The first is for the probability that there is an open path connecting the two boundaries of an annulus of inner radius $r$ and outer radius $R$. The second is for the probability that there are both open and closed paths connecting the two annulus boundaries. These two results were predicted by Cardy based on non-rigorous Coulomb gas arguments. Our third result gives the probability that there are two disjoint open paths connecting the two boundaries. Its leading asymptotic as $r/R\to 0$ is captured by the so-called backbone exponent, a transcendental number recently determined by Nolin, Qian and two of the authors. This exponent is the unique real root to the equation $\frac{\sqrt{36 x +3}}{4} + \sin (\frac{2 π\sqrt{12 x +1}}{3} ) =0$, other than $-\frac{1}{12}$ and $\frac{1}{4}$. Besides these three real roots, this equation has countably many complex roots. Our third result shows that these roots appear exactly as exponents of the subleading terms in the crossing formula. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Expanding the same crossing probability as $r/R\to 1$, we obtain a series with logarithmic corrections at every order, suggesting that the backbone exponent is related to a logarithmic boundary CFT. Our proofs are based on the coupling between SLE curves and Liouville quantum gravity (LQG). The key is to encode the annulus crossing probabilities by the random moduli of certain LQG surfaces with annular topology, whose law can be extracted from the dependence of the LQG annuli partition function on their boundary lengths.

math.PR↗