台大校區機率演講公告Probability Seminars on NTU Campus

In this talk, following the talk on Nov. 5, I will further review the asymptotic behavior of empirical spectral distribution (ESD) of the sparse random matrices. I will also present my recent work and some simulations.

In this talk, I will review the foundational framework for analyzing the convergence of large-dimensional random matrices' empirical spectral distribution (ESD) as matrix dimensions grow. If time permits, I will also present some simulation results and discuss my recent work.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\\chi_z=\\sum_{n=0}^{\\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\\gamma$ such that $\\chi_z$ diverges as $(z_c-z)^{-\\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\\le4$, while it has been proved that $\\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\\gamma=1$ for $d>4$.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\\chi_z=\\sum_{n=0}^{\\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\\gamma$ such that $\\chi_z$ diverges as $(z_c-z)^{-\\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\\le4$, while it has been proved that $\\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\\gamma=1$ for $d>4$.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. Despite this simple definition, many of the most basic questions are difficult to resolve in a mathematically rigorous fashion. However, since in high dimension the principal effect of the self-avoidance constraint become relatively weak, a lot of results have been proved, compared to low dimensional case. For example, introducing a parameter $z$, we consider a generating function $\\chi_z=\\sum_{n=0}^{\\infty}C_nz^n$ where $C_n$ is the number of self-avoiding walks starting from origin. It is believed that there exists dimension-dependent exponent $\\gamma$ such that $\\chi_z$ diverges as $(z_c-z)^{-\\gamma}$ when $z$ is getting close to the radius of convergence $z_c$.

Even the existence of such critical exponent is still unknown when $d\\le4$, while it has been proved that $\\gamma=1$ when $d>4$. The lace expansion is known as one of the most powerful tool to show the 'mean-field behavior' of statistical mechanical models in high dimension, which played an important role to prove $\\gamma=1$ for $d>4$. In this seminar, we will explore how the lace expansion is derived and its mathematical foundation, and we will discuss how the lace expansion can be used to establish mean-field type results and specifically to prove that $\\gamma=1$ for $d>4$.

The self-avoiding walk (SAW) is a model defined by adding self-avoidance interaction to the random walk. In other words, each path does not visit the same vertex on a graph more than once. We consider the connectivity function $c_n(x)$ defined by the number of $n$-step SAWs from the origin $o$ to a vertex $x$. It is known that the spread-out SAW with finite-range interactions enjoys the central limit theorem [van der Hofstad and Slade (2002) PTRF][van der Hofstad and Slade (2003) AAM]. Taking an average on a ball, they also proved a certain type of a local limit theorem for $c_n(x)$. For the spread-out SAW with long-range interactions whose one-step distribution has heavy tails, the power-law decay of the two-point function $G_p(x) = \\sum_{x \\in \\mathbb{Z}^d} c_n(x) p^n$ was shown in [Chen and Sakai (2015) AOP][Chen and Sakai (2019) CMP].

In this talk, I will explain an attempt to prove a local limit theorem for the spread-out long-range SAW in the original sense. Our motivations come from combining the results of the previous researches. I will show two different strategies. The lace expansion gives a certain type of a recurrence relation for the sequence $\\{c_n(x)\\}_{n=1}^{\\infty}$. The first one is based on the analogous approach with [Chen and Sakai (2019) CMP] in which we substitute the recurrence relation into $\\{c_i(x)\\}_{i=1}^{n-1}$. The second one is based on the inductive approach [van der Hofstad and Slade (2002) PTRF] extended to the long-range model in which we assume an upper bound on $c_n(x)$ for $1 \\leq m \\leq n$ and prove it for $2 \\leq m \\leq n+1$. I will report the current progress of our attempts using these approaches.

This talk is joint work with Lung-Chi Chen (National Chengchi University) and Yuki Chino (National Yang-Ming Chiao-Tung University).