15 June 2026, Volume 42 Issue 6

    

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  Preface of the Special Issue on Discrete Probability and Mathematical Physics

  Zhen-Qing Chen, Jian Ding, Fuzhou Gong, Zhiming Ma, Zhan Shi, Hao Wu

  Acta Mathematica Sinica. 2026, 42(6): 1431-1431. https://doi.org/10.1007/s10114-026-5640-5

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  The Intermediate Level-sets of the Four-dimensional Membrane Model

  Xinyi Li, Runsheng Liu

  Acta Mathematica Sinica. 2026, 42(6): 1432-1456. https://doi.org/10.1007/s10114-026-4341-4

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  In this paper, we consider the discrete membrane model in four dimensions. We confirm the existence of the scaling limit of the intermediate (i.e., a multiple of the expected maximum) level-sets of the model, and show that it is equal in law to a tilted version of the sub-critical Gaussian multiplicative chaos (GMC) measure of the continuum membrane model.
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  Infinite Differentiability of the Free Energy for a Derrida-Retaux System

  Xinxing Chen

  Acta Mathematica Sinica. 2026, 42(6): 1457-1480. https://doi.org/10.1007/s10114-026-4249-z

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  We consider a recursive system which was introduced by Derrida and Retaux (J. Stat. Phys., ${\bf 156}$, 268-290 (2014)) as a toy model to study the depinning transition in presence of disorder. Derrida and Retaux predicted the free energy $F_\infty(p)$ of the system exhibit quite an unusual physical phenomenon which is an infinite order phase transition. Hu and Shi (J. Stat. Phys., ${\bf 172}$, 718-741 (2018)) studied a special situation and obtained other behavior of the free energy, while insisted on $p=p_c$ being an essential singularity. Recently, Chen, Dagard, Derrida, Hu, Lifshits and Shi (Ann. Probab., ${\bf 49}$, 637-670 (2021)) confirmed the Derrida-Retaux conjecture under suitable integrability condition. However, from a mathematical point of view, it is still unknown whether the free energy is infinitely differentiable at the critical point. So we continue to study the infinite differentiability of the free energy in this paper.
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  The Behavior of Observables in Renormalization

  Kaiyuan Cui, Fuzhou Gong

  Acta Mathematica Sinica. 2026, 42(6): 1481-1508. https://doi.org/10.1007/s10114-026-4322-7

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  In this paper, we introduce novel reference observables for the sake of establishing a scaling formula in the renormalization group (RG) equation. Firstly, using the transfer matrix method, we calculate the two point observables of the one dimensional (1D) Ising model without an external field under general boundary conditions. The results suggest that the two point observables decay exponentially except at the critical point. Corresponding to the RG procedure underlying the correlation function, we establish a similar procedure for new observables, which is consistent with the findings in physics. Secondly, from a dynamic perspective, we construct a random system via the stochastic quantization method. We calculate the new observables of this random system under the initial distribution satisfying the Dobrushin-Lanford-Ruelle (DLR) equations. Furthermore, we formulate a new renormalization scaling formula with respect to the two point observables. Finally, these results can be extended to any finite point observables, and are independent of the choice of system parameters.
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  Joint Scaling Limit of a Bipolar-Oriented Triangulation and Its Dual in the Peanosphere Sense

  Ewain Gwynne, Nina Holden, Xin Sun

  Acta Mathematica Sinica. 2026, 42(6): 1509-1554. https://doi.org/10.1007/s10114-026-4359-7

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  Kenyon, Miller, Sheffield, and Wilson (2015) showed how to encode a random bipolar-oriented planar map by means of a random walk with a certain step size distribution. Using this encoding together with the mating-of-trees construction of Liouville quantum gravity (LQG) due to Duplantier, Miller, and Sheffield (2014), they proved that random bipolar-oriented planar maps converge in the scaling limit to a $\sqrt{\frac{4}{3}}$-LQG surface decorated by an independent SLE$_{12}$ in the peanosphere sense, meaning that the height functions of a particular pair of trees on the maps converge in the scaling limit to the correlated planar Brownian motion which encodes the SLE-decorated LQG surface. We improve this convergence result by proving that the pair of height functions for an infinite-volume random bipolar-oriented triangulation and the pair of height functions for its dual map converge jointly in law in the scaling limit to the two planar Brownian motions which encode the same $\sqrt{\frac{4}{3}}$-LQG surface decorated by both an SLE$_{12}$ curve and the ``dual'' SLE$_{12}$ curve which travels in a direction perpendicular (in the sense of imaginary geometry) to the original curve. This confirms a conjecture of Kenyon, Miller, Sheffield, and Wilson (2015). Our paper is the starting point of recent works connecting LQG and random permutons such as the Baxter permuton.
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  Large Deviations for the Extremal Eigenvalues of Ginibre Ensembles

  Yuanyuan Xu, Qiang Zeng

  Acta Mathematica Sinica. 2026, 42(6): 1555-1571. https://doi.org/10.1007/s10114-026-4382-8

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  We establish large deviation principles for the extremal eigenvalues of the Ginibre ensembles with good rate functions. In contrast to the typical estimates for the extremal eigenvalues, the large deviations for the real Ginibre ensemble come from the eigenvalues lying on the real line. Moreover, we also derive deviation estimates for the second leading term in the asymptotic expansion of the extremal eigenvalues. These polynomially small deviation estimates are universal for any i.i.d. matrices under a mild moment condition.
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  Size Distribution of Clusters in Site-percolation on Random Recursive Tree

  Chenlin Gu, Linglong Yuan

  Acta Mathematica Sinica. 2026, 42(6): 1572-1594. https://doi.org/10.1007/s10114-026-4437-x

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  We prove rigorously several results about the site-percolation on random recursive trees, observed in the previous work by Kalay and Ben-Naim (J. Phys. A., 2015). For a random recursive tree of size $n$, let every site have probability ${p \in (0,1)}$ to remain and with probability $(1-p)$ to be removed. As $n\to\infty,$ we show that the proportion of the remaining clusters of size $k$ is of order $k^{-1-\frac{1}{p}}$, resulting in a Yule-Simon distribution; the largest cluster size is of order $n^{p}$, and admits a non-trivial scaling limit. The proofs are based on the embedding of this model in the multi-type branching processes, and a coupling with the bond-percolation on random recursive trees.
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  Number Rigid Determinantal Point Processes Induced by Generalized Cantor Sets

  Zhaofeng Lin, Yanqi Qiu, Kai Wang

  Acta Mathematica Sinica. 2026, 42(6): 1595-1608. https://doi.org/10.1007/s10114-026-4384-6

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  We consider the Ghosh-Peres number rigidity of translation-invariant determinantal point processes on the real line $\mathbb{R}$, whose correlation kernels are induced by the Fourier transform of the indicators of generalized Cantor sets in the unit interval. Our main results show that for any given $\theta\in(0,1)$, there exists a generalized Cantor set with Lebesgue measure $\theta$, such that the corresponding determinantal point process is Ghosh-Peres number rigid.
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  A Central Limit Theorem for Square Ice

  Wei Wu

  Acta Mathematica Sinica. 2026, 42(6): 1609-1620. https://doi.org/10.1007/s10114-026-4385-5

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  We prove that the height function associated with the uniform six-vertex model (or equivalently, the uniform homomorphism height function from $\mathbb{Z}^2$ to $\mathbb{Z}$) satisfies a central limit theorem, upon some logarithmic rescaling.
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  Large Deviations for the Maximum of a Reducible Two-type Branching Brownian Motion

  Hui He

  Acta Mathematica Sinica. 2026, 42(6): 1621-1638. https://doi.org/10.1007/s10114-026-4393-5

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  We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Belloum and Mallein showed that the maximum position $M_t$ of all particles alive at time $t$, suitably centered by a deterministic function $m_t$, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as $t\rightarrow\infty$, \[ \mathbb{P}(M_t\geq \theta m_t),\quad \theta>1. \] We shall show that the decay rate function exhibits phase transitions depending on certain relations between $\theta$, the variance of the underlying Brownian motion and the branching rate.
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  Boundedness of Discounted Tree Sums

  Elie Aïdékon, Yueyun Hu, Zhan Shi

  Acta Mathematica Sinica. 2026, 42(6): 1639-1661. https://doi.org/10.1007/s10114-026-4459-4

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  Let $(V(u),\, u\in T)$ be a (supercritical) branching random walk and $(\eta_u,\,u\in T)$ be marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047-1110 (2005)], for each ray $\xi$ of the tree, we associate the discounted tree sum $D(\xi)$ which is the sum of the ${\rm e}^{-V(u)}\eta_u$ taken along the ray. The paper deals with the finiteness of $\sup_\xi D(\xi)$. To this end, we study the extreme behaviour of the local time processes of the paths $(V(u),\,u\in \xi)$. It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047-1110 (2005)]. We also present several open questions.
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  Large Deviations at the Origin of Random Walk in Random Environment

  Alexander Drewitz, Alejandro Ramírez, Santiago Saglietti, Zhicheng Zheng

  Acta Mathematica Sinica. 2026, 42(6): 1662-1684. https://doi.org/10.1007/s10114-026-4531-0

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  We consider a random walk in an independent and identically distributed (i.i.d.) random environment on $\mathbb Z^d$ and study properties of its large deviation rate function at the origin. It was proved by Comets, Gantert and Zeitouni in dimension $d=1$ in 1999 and later by Varadhan in dimensions $d\ge 2$ in 2003 that, for uniformly elliptic i.i.d. random environments, the quenched and the averaged large deviation rate functions coincide at the origin. Here we provide a description of an atypical event realizing the correct quenched large deviation rate in the nestling and marginally nestling setting: the random walk seeks regions of space where the environment emulates the element in the convex hull of the support of the law of the environment at a site which minimizes the rate function. Periodic environments play a natural role in this description.
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  Increasing Paths on N-ary Trees

  Xinxin Chen

  Acta Mathematica Sinica. 2026, 42(6): 1685-1712. https://doi.org/10.1007/s10114-026-5033-9

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  Consider a rooted $N$-ary tree. To every vertex of this tree, we attach an i.i.d. continuous random variable. A vertex is called accessible if along its ancestral line, the attached random variables are increasing. We keep accessible vertices and kill all the others. For any positive constant $\alpha$, we describe the asymptotic behaviors of the population at the $\alpha N$-th generation as $N$ goes to infinity. We also study the criticality of the survival probability at the $({e}N-\frac{3}{2}\log N)$-th generation in this paper.
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  A Moment Bound on Bulk/Boundary Quotients of Gaussian Multiplicative Cascades

  Yichao Huang, Youtao Liu

  Acta Mathematica Sinica. 2026, 42(6): 1713-1726. https://doi.org/10.1007/s10114-026-5036-6

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  We initiate the study of bulk/boundary quotients of Gaussian multiplicative cascades measures, for which we establish preliminary joint moment bounds. This is a preliminary result towards studying the bulk/boundary quotients of Gaussian multiplicative chaos measures coming from the theory of boundary Liouville conformal field theory.