First Passage 首达

MSc dissertation: first passage in confined random walks

The MSc in Engineering Mathematics at Bristol (2024–25) finished with a Distinction and a dissertation titled ‘First-Passage Phenomena in Confined Random Walks: Finite-Size Effects, Exact Solutions, and Spatial Heterogeneity’. It asked how long a random walker takes to first reach a target on one-, two- and three-dimensional lattices, with and without defects, and set exact solutions against simulation. The code is public. It is the first place the walker got trapped in a box — and the question has not let go since.

PhD in Engineering Mathematics, University of Bristol

The PhD began on 3 November 2025 in the School of Engineering Mathematics and Technology (SEMT) at the University of Bristol. The subject is first-passage time: how long a random walker takes to reach a target for the first time, and why the distribution of that time can split into two peaks. Over the first year the work has broadened to span stochastic processes and the evaluation of AI agents. The random walks moving across this page are the same kind of object the thesis studies.

PhD researcher

First PhD code: a ring with defects

Within four weeks of starting, a first prototype was specified and built: a Python simulation of a random walk on a one-dimensional ring with defects, with analysis scripts, parameter scans and a small web visualiser. One configuration already hinted at a two-humped first-passage histogram. It was a sketch rather than a result, and it was archived once the exact methods arrived — but it is the baby picture of the whole research line.

Computing on the Isambard supercomputers

From December 2025 the heavy computation ran on Bristol's national supercomputers, as part of a university project on random walkers in disordered environments. Command-line automation syncs code, submits SLURM jobs and fetches results. Isambard-AI carried the continuum-bridge production for the encounter paper — lattice ladders refined up to n = 33 and Brownian dynamics with 2×10⁷ walkers per release position — and Isambard 3 ran the large-scale checks for the fixed-budget paper in September 2026, including a census of 2×10⁹ walkers.

Lazy rings with a shortcut: the bilingual reports

The ring studies were consolidated into bilingual reports: a lazy ring with a one-way shortcut, two neighbour kernels, scans over shortcut strength β and ring size N, Monte Carlo trajectory classes and tail diagnostics. The long-time tail turned out to be set by the spectral radius of the transient chain, and β* = 0.01 was the smallest shortcut strength at which both kernels meet the double-peak criterion. A suspicious plateau in one figure was traced to a truncation in the numerical inversion and fixed — the first of many bugs caught by refusing to trust a pretty curve.

From rings to two-dimensional lattices

In 2026 the question moved from rings to two-dimensional lattices — periodic, rectangular and reflecting domains, a membrane near the target, two competing targets, local bias fields and two-walker encounters. Of the 27 reports in the public research atlas, 13 belong to this lattice family. One finding: a membrane's permeability mainly adds to the time spent after the crossing, which is why some geometries keep a clear double peak even with minimal bias.

A public, bilingual research atlas

On 25–26 February 2026 the first-passage line got its own public website: a static research atlas in English and Chinese, with a continuous ‘book’ of eight chapters, a page for every report, a theory section, interactive figures and typeset mathematics. It maps 27 reports and tracks 130 claims back to their evidence. The repository behind it is organised ‘agent-first’ — one command-line tool, guardrail tests and a machine-readable report registry — so that people and AI agents can work on it in the same way.

BAMC 2026, Norwich

Attended the British Applied Mathematics Colloquium 2026 at the University of East Anglia in Norwich, which opened on 30 March 2026 — the first national applied-mathematics meeting of the PhD, and a first look at where first-passage questions sit within the wider field.

‘Hidden Routes in First-Passage Time’

A ten-minute research talk for a broad audience of SEMT PhD students, built as a custom web slide deck inside the research atlas, with a live random-walk animation and figures regenerated from the research code. Its one-line thesis: a single target on a ring can still produce a fast branch, a valley and a delayed branch once one directed shortcut is present. It was scheduled for the SEMT Ideas Exchange Day on 15 April 2026; the deck and the script are public in both languages.

Three ingredients of a double peak

A 12-page brief pulled the year's cases together — ring shortcuts, two-target lattices and exact two-walker encounter searches — and argued that a visible double peak needs three things at once: a balance between the masses of the fast and slow channels, well-separated timescales, and a valley deep enough to resolve. A second talk page on multi-timescale encounters, with a live two-dimensional hopping demo, joined the atlas on 31 May 2026.

The term that cancels

For a lazy ring with an antipodal shortcut, a ‘t·αᵗ’ term was suspected in the long-time tail of the first-passage distribution. It is not there. Three independent arguments show that its coefficient is identically zero: a partial-fraction sum rule; an account of the term as an artefact of a boundary slice in the generating-function inversion; and a spectral argument — the transient matrix is symmetric, hence diagonalisable, so terms like t·λᵗ cannot occur for any parameters. Every identity was checked in 50-digit arithmetic and with exact rational kernels.

Why a second peak is born — and dies

The ring's double peak became a threshold theory. A directed shortcut into an absorbing target behaves like a rank-one killing defect; in the diffusive limit the problem is Brownian motion on an interval with an interior sink of strength b. The first-passage density is then a signed mixture of spectral modes, and the late peak disappears in a saddle-node fold at b = b_c(θ) — about 3.0764 in the symmetric case — with fold exponents 1/2 and 3/2 and at least three modes required. The result was cross-checked five ways: exact lattice residues, Monte Carlo with 4×10⁵ walkers, an exact channel decomposition, Brownian dynamics, and finite two-dimensional lattices.

Working paper: spectral diagnostics for encounter–reaction models

A single-author working paper on when two diffusing particles that react on contact produce more than one reaction-time peak in finite models. Its tools include a reduction onto the reaction support, a sign-count gate (m peaks need at least 2m − 1 sign changes among the residues), an exact response to redistributing a fixed reactivity budget, and a continuum bridge built from refined two-dimensional lattice ladders and off-lattice Brownian dynamics. The headline conclusion: folds seen on coarse grids escape to the boundary of the admissible region in the continuum. Related work in progress looks at encounters with several reactive zones.

An interactive course for one's own paper

To understand the fold paper from the ground up, a single-file offline course was built around it: ten chapters that start from undergraduate analysis and linear algebra, build double-peak intuition from walker histograms, show how negative mode weights create peaks, and end with a draggable b-slider that breaks the saddle-node live. Every chapter has self-tests. It is a study tool rather than course material — and the seed of the explainer in this chapter.

46 theorems, checked by machine

The exact algebraic layer of the fold paper — Chebyshev product identities, Green's-function columns, the rank-one determinant, the closed-form splitting probability, jump conditions, the normal-form prefactor 4√2/3 and the minimal three-mode theorem — was formalised in Lean 4 with mathlib: 46 theorems across seven modules, with no ‘sorry’, depending only on Lean's three standard axioms. Numerical constants and analytic limits were deliberately left out of scope and verified numerically instead.

Geometry-controlled folds — submitted to Physical Review E

‘Geometry-controlled folds of first-passage-time bimodality under localized absorption’ was submitted to Physical Review E as a single-author regular article on 30 July 2026. It is the first paper of the PhD and the capstone of the double-peak line: the geometry of a localized absorber decides whether a second arrival peak exists at all, and the peak is lost through a saddle-node fold whose location can be computed exactly. Every figure on this site that shows the effect is generated by our own simulations, not taken from the manuscript.

Programming the number of peaks

For a reacting Brownian pair swept past stripes of catalyst, a first theorem (August 2026) showed that a small enough budget split across m stripes yields exactly m reaction-time peaks. A campaign of about 325 million simulated walkers tested it: all 48 cells of the two-stripe phase diagram gave exactly two peaks, a five-peak demonstration put five peaks at their designed times, and a three-dimensional spot check passed unchanged. An explicit bound on the admissible budget was then sharpened by 692 orders of magnitude, and the finite-dimensional core was formalised in Lean 4 (138 audited declarations, no ‘sorry’).

prescribed-reaction-time-modes: a public reproduction archive

Everything behind the fixed-budget paper is public and checkable: simulation, analysis and plotting code, machine-readable numerical records, interval-arithmetic certificates, the Lean 4 sources and the Isambard 3 job records, each tied to SHA-256 manifests. Five releases went out between 8 and 26 September 2026 as the paper was rebuilt — about 1,466 files in the latest, including 100 Python scripts, 27 Lean files and 1,198 JSON records. Anyone can re-run the numbers without asking.

Programmable multimodality — manuscript in preparation

‘Programmable multimodality and saddle-node loss of reaction-time peaks under a fixed catalyst budget’ is a single-author manuscript in preparation. A fixed amount of static catalyst, split into thin stripes along the path of a reacting Brownian pair, programs the reaction-time density: at weak noise each stripe removes a fixed fraction of the arriving pairs — a stick-breaking law — so there is exactly one peak per stripe, and designs with prescribed peak times and weights exist, are locally unique and can be found by Newton's method. Fixed-width stripes can instead lose a late peak through a saddle-node bifurcation, certified by interval arithmetic in test cases. The finite-dimensional and calculus core is checked in Lean 4; the stochastic parts are proved by hand.