arXiv subjects

Tsung-Hsuan Tsai

Publications and source records attributed to Tsung-Hsuan Tsai.

6 recordsLinked to original sources

arXiv2024

Parallel geodesics and minimal stable length of random groups

We show that for any pair of long enough parallel geodesics in a random group $G_\ell(m,d)$ with $m$ generators at density $d<1/6$, there is a van Kampen diagram having only one layer of faces. Using this result, we give an upper bound, depending only on $d$, of the number of pairwise parallel geodesics in $G_\ell(m,d)$ when $d<1/6$. As an application, we show that the minimal stable length of $G_\ell(m,d)$ at $d<1/6$ is exactly $1$.

math.GR↗

arXiv2022

Phase transition for the existence of van Kampen 2-complexes in random groups

Gromov showed that (1993) with high probability, every bounded and reduced van Kampen diagram $D$ of a random group at density $d$ satisfies the isoperimetric inequality $|\partial D|\geq (1-2d-s)|D|\ell$. In this article, we adapt Gruber-Mackay's prove for random triangular groups, showing a non-reduced 2-complex version of this inequality. Moreover, for any 2-complex $Y$ of a given geometric form, we exhibit a phase transition: we give explicitly a critical density $d_c$ depending only on $Y$ such that, in a random group at density $d$, if $d d_c$ then there exists reduced van Kampen 2-complexes of the form $Y$. As an application, we show a phase transition for the $C(p)$ small-cancellation condition: for a random group at density $d$, if $d<1/(p+1)$ then it satisfies $C(p)$; while if $d>1/(p+1)$ then it does not satisfy $C(p)$.

math.GR↗

arXiv2021

Density of random subsets and applications to group theory

Developing an idea of M. Gromov, we study the intersection formula for random subsets with density. The \textit{density} of a subset $A$ in a finite set $E$ is defined by $dens A := \log_{|E|}(|A|)$. The aim of this article is to give a precise meaning of Gromov's \textit{intersection formula}: "Random subsets" $A$ and $B$ of a finite set $E$ satisfy $dens (A\cap B) = dens A + dens B -1$. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density $λ/2$ for any $0<λ<1$, characterizing the $C'(λ)$-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii from density $0$ to density $0\leq d<\frac{1}{120m^2\ln(2m)}$.

math.GR↗

arXiv2016

On the linearity of Regge trajectory at large transfer energy

In this note we consider the four-point function of identical scalar operators in its Mellin representation. For CFTs, taking the large scaling dimension limit of the Mellin amplitude yields the flat-space S-matrix. Using this correspondence, and assuming that the Regge-limit of the Mellin amplitude vanishes, we prove that the resulting flat space S-matrix must have string-like linear Regge trajectory in the limit |s|>>|t|>>1. We also show that the Regge limit of Mellin amplitude is captured by the three-point coefficient of two scalars and one double trace operator in the large-spin limit. Using this, we can see that vanishing Regge behaviour occurs at small t'Hooft limit.

hep-th↗

arXiv2015

On Higgs Branch Localization of Seiberg-Witten Theories on Ellipsoid

In this note, we consider so-called "Higgs Branch Localization" for four dimensional N=2 field theories on 4d ellipsoid. We find a new set of saddle point equations arising from additional Higgs branch deformation term, whose solutions include both Higgs branch and BPS instanton-vortex mixed configurations. By evaluating the contour integral, we also demonstrate the ellipsoid partition almost factorizes into purely b and 1/b dependent parts, using SQCD as an explicit example. We identify various factorized parts with the ellipsoid partition function of two dimensional N=(2,2) SQCDA, which is precisely the vortex world volume theory. We also give physical interpretation for the non-factorizable parts and discuss future directions.

hep-th↗

arXiv2013

Random triangular groups at density 1/3

Let Γ(n,p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p <= c/n^2, then a.a.s. Γ(n,p) is free and if p >= C log n/n^2 then a.a.s. Γ(n,p) has Kazhdan's property (T). Furthermore, we show that there exist constants C',c' > 0 such that if C'/n^2 <= p <= c' log n/n^2, then a.a.s. Γ(n,p) is neither free nor has Kazhdan's property (T).

math.GR↗