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1. Universality and phase transitions in low moments of secular coefficients of critical holomorphic multiplicative chaos

   https://doi.org/10.1007/s00440-025-01443-z  

   Gu, Haotian; Zhang, Zhenyuan (November 2025, Probability Theory and Related Fields)

   Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0 $E\left[\right|A_N{|}^{2q}],0<q⩽1$of secular coefficients$$A_N$$ $A_N$of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$ $z^N$in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$ $exp({\sum }_{k=1}^{\infty }{X}_k{z}^k/\sqrt{k})$, where$$\{X_k\}_{k\geqslant 1}$$ ${\left\{X_k\right\}}_{k⩾1}$are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper’s remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each$$X_k$$ $X_k$is standard complex Gaussian,$$A_N$$ $A_N$features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$ $E\left[\right|A_N{|}^2]=1$and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$ $E\left[\right|A_N{|}^{2q}]\asymp {(logN)}^{-q/2}$for fixed$$q\in (0,1)$$ $q\in (0,1)$as$$N\rightarrow \infty $$ $N\to \infty$. We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$ $E\left[e^{\gamma |X_k|}\right]<\infty$for some$$\gamma >2q$$ $\gamma >2q$. As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$ $A_N{(log(1+N))}^{1/4}$, generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of$$\mathbb {E}[|A_N|^{2q}]$$ $E\left[\right|A_N{|}^{2q}]$for$$|X_k|$$ $|X_k|$following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper’s robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of$$A_N$$ $A_N$. 

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2. The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

   https://doi.org/10.1007/s00526-021-01938-2  

   Banerjee, Agnid; Danielli, Donatella; Garofalo, Nicola; Petrosyan, Arshak (April 2021, Calculus of Variations and Partial Differential Equations)

   Abstract We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight$$|y|^a$$ ${\left|y\right|}^a$for$$a \in (-1,1)$$ $a\in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator$$(\partial _t - \Delta _x)^s$$ ${({\partial }_t-{\Delta }_x)}^s$for$$s \in (0,1)$$ $s\in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ $a=0$). 

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3. On a Traveling Salesman Problem for Points in the Unit Cube

   https://doi.org/10.1007/s00453-024-01257-w  

   Balogh, József; Clemen, Felix Christian; Dumitrescu, Adrian (September 2024, Algorithmica)

   Abstract LetXbe ann-element point set in thek-dimensional unit cube$$[0,1]^k$$ ${[0,1]}^k$where$$k \ge 2$$ $k\ge 2$. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour)$$x_1, x_2, \ldots , x_n$$ $x_1,x_2,\dots ,x_n$through thenpoints, such that$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} \le c_k$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}\le c_k$, where$$|x-y|$$ $|x-y|$is the Euclidean distance betweenxandy, and$$c_k$$ $c_k$is an absolute constant that depends only onk, where$$x_{n+1} \equiv x_1$$ $x_{n+1}\equiv x_1$. From the other direction, for every$$k \ge 2$$ $k\ge 2$and$$n \ge 2$$ $n\ge 2$, there existnpoints in$$[0,1]^k$$ ${[0,1]}^k$, such that their shortest tour satisfies$$\left( \sum _{i=1}^n |x_i - x_{i+1}|^k \right) ^{1/k} = 2^{1/k} \cdot \sqrt{k}$$ ${\left({\sum }_{i=1}^n,{|x_i-x_{i+1}|}^k\right)}^{1/k}=2^{1/k}·\sqrt{k}$. For the plane, the best constant is$$c_2=2$$ $c_2=2$and this is the only exact value known. Bollobás and Meir showed that one can take$$c_k = 9 \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=9{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$for every$$k \ge 3$$ $k\ge 3$and conjectured that the best constant is$$c_k = 2^{1/k} \cdot \sqrt{k}$$ $c_k=2^{1/k}·\sqrt{k}$, for every$$k \ge 2$$ $k\ge 2$. Here we significantly improve the upper bound and show that one can take$$c_k = 3 \sqrt{5} \left( \frac{2}{3} \right) ^{1/k} \cdot \sqrt{k}$$ $c_k=3\sqrt{5}{\left(\frac{2}{3}\right)}^{1/k}·\sqrt{k}$or$$c_k = 2.91 \sqrt{k} \ (1+o_k(1))$$ $c_k=2.91\sqrt{k}(1+o_k\left(1\right))$. Our bounds are constructive. We also show that$$c_3 \ge 2^{7/6}$$ $c_3\ge 2^{7/6}$, which disproves the conjecture for$$k=3$$ $k=3$. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed. 

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4. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa (April 2022, Mathematische Annalen)

   Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ $a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$ $a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$ $k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$ $a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$ $j=1,\cdots ,m$and that every$$m\times m$$ $m\times m$minor of the$$m\times k$$ $m\times k$matrix$$(a_{j,i})_{j,i}$$ ${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$ $\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ $x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$ $\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$ $(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ $x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 

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5. A Refutation of the Pach-Tardos Conjecture for 0–1 Matrices

   https://doi.org/10.1007/s00493-025-00187-7  

   Pettie, Seth; Tardos, Gábor (December 2025, Combinatorica)

   Abstract The theory of forbidden 0–1 matrices generalizes Turán-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parametertwin width. The foremost open problem in this area is to resolve thePach-Tardos conjecturefrom 2005, which states that if a forbidden pattern$$P\in \{0,1\}^{k\times l}$$ $P\in {\{0,1\}}^{k\times l}$isacyclic, meaning it is the bipartite incidence matrix of a forest, then$$\operatorname {Ex}(P,n) = O(n\log ^{C_P} n)$$ $Ex(P,n)=O\left(n{log}^{C_P}n\right)$, where$$\operatorname {Ex}(P,n)$$ $Ex(P,n)$is the maximum number of 1s in aP-free$$n\times n$$ $n\times n$0–1 matrix and$$C_P$$ $C_P$is a constant depending only onP. This conjecture has been confirmed on many small patterns, specifically allPwith weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that$$\operatorname {Ex}(S_0,n),\operatorname {Ex}(S_1,n) \ge n2^{\Omega (\sqrt{\log n})}$$ $Ex(S_0,n),Ex(S_1,n)\ge n{2}^{\Omega \left(\sqrt{logn}\right)}$, where$$S_0,S_1$$ $S_0,S_1$are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class ofalternatingpatterns$$(P_t)$$ $\left(P_t\right)$, specifically that for every$$t\ge 2$$ $t\ge 2$,$$\operatorname {Ex}(P_t,n)=\Theta (n(\log n/\log \log n)^t)$$ $Ex(P_t,n)=\Theta \left(n{(logn/loglogn)}^t\right)$. This is the first proof of an asymptotically sharp bound that is$$\omega (n\log n)$$ $\omega (nlogn)$. 

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