We consider first-passage percolation (FPP) on the book graph with multiple pages, where upper-half planes are glued along the common axis. FPP was introduced by Hammersley and Welsh in 1965 as a model of fluid flow through a random medium. In the model, a non-negative random variable \( t_e \) is assigned on each edge of the graph, independently of the others. The passage time of a path is defined as the sum of the \( t_e \)'s over edges traversed by the path.
Our interest is in the infimum of the passage times over all finite paths from \( o \) to \( ne_1 \), which is defined by \( T(0,ne_1) \). In this talk, we prove that when the number of pages of the book graph is sufficiently large, the variance of \( T(0,ne_1) \) is of order \( n \), which is markedly different from the conjectured behavior on two-dimentional integer lattice, where the variance is of order \( n^{2/3} \). This talk is based on joint work with Tzu-Han Chou (NUS) and Wai-Kit Lam (NTU).
◆感染症対策のため、風邪症状（咳が出る、体がだるい）もしくは
発熱（37.5℃以上、または平熱より1.0℃以上高い熱）のある方は
参加をご遠慮下さいますようお願いいたします。
◆「最初の2/3程度は非専門家（専門外の数学者）向けの講演とすること」を
お願いしております。
◆多数の教員および学生の方の参加をお待ちしています。
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Date : 2026.4.22 (Wed) 16:45-17:45 (16:15- tea in 2nd floor common room)
Place : Rm110, RIMS, Kyoto University
Speaker : Noe Kawamoto (Kyoto University)
Title : Linear Variance of First-Passage Percolation on the Book Graph
Abstract:
We consider first-passage percolation (FPP) on the book graph with
multiple pages, where upper-half planes are glued along the common axis.
FPP was introduced by Hammersley and Welsh in 1965 as a model of fluid flow
through a random medium. In the model, a non-negative random variable $t_e$
is assigned on each edge of the graph, independently of the others. The
passage
time of a path is defined as the sum of the $t_e$'s over edges traversed by
the path. Our interest is in the infimum of the passage times over all
finite
paths from $o$ to $ne_1$, which is defined by $T(0,ne_1)$. In this talk,
we prove that when the number of pages of the book graph is sufficiently
large, the variance of $T(0,ne_1)$ is of order $n$, which is markedly
different
from the conjectured behavior on two-dimentional integer lattice, where the
variance is of order $n^{2/3}$. This talk is based on joint work with
Tzu-Han
Chou (NUS) and Wai-Kit Lam (NTU).
Language: Japanese (with English slides or board)
◆As a measure against infectious diseases, please refrain from
attending the talk if you have a cold (have a cough, feel heavy)
or a fever (above 37.5℃, or at least 1.0℃ higher than normal).
◆We have requested the speaker to ``keep the first 2/3 of the talk
accessible to non-experts (mathematicians in different fields)''.
◆We look forward to active participation by faculty members and
students.