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The Drunkard's Walk by Leonard Mlodinow, book cover

The Drunkard's Walk Summary

by Leonard Mlodinow

3.95 Goodreads · 25k ratings235 pages15-min summary

Mlodinow shows how our lives are a high-stakes game of chance, revealing the hidden mathematical forces behind everything from lottery wins to superstardom, and why we often misunderstand them.

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By BookBrief EditorialPublished February 2026How we write summaries

Core Idea

The Drunkard's Walk explains how randomness affects our lives and the world, often in ways we mistake for skill, intent, or clear cause-and-effect. Mlodinow argues that from financial markets to medical diagnoses, and from personal success to scientific discovery, chance plays a bigger role than people usually admit. By exploring ideas like regression to the mean, the law of large numbers, and common probability errors, the book offers a way to understand uncertainty. The main idea is that understanding how randomness works—instead of trying to get rid of it—is important for making better decisions, interpreting events more accurately, and living in a world shaped by statistics. It challenges our natural human desire to find patterns and assign reasons, pushing readers to think probabilistically to avoid common mental errors and improve their predictions.
Difficulty
Medium
Read it if
You want to understand the pervasive role of randomness and probability in everyday life, decision-making, and the interpretation of events, and how to avoid common cognitive biases related to chance.
Skip it if
You are looking for a prescriptive self-help guide or a book that offers deterministic solutions to complex problems, or if you already have a strong background in statistics and probability theory.

Key Takeaways from The Drunkard's Walk

The Illusion of Control

We systematically overestimate our skill and underestimate the role of luck.

People tend to credit outcomes, especially good ones, to skill and deliberate action rather than to chance. This 'illusion of control' makes us believe we can influence events largely beyond our reach. Mlodinow shows how this mental error affects everything from financial markets to sports, causing us to misinterpret success and failure. We look for patterns where there are none, creating stories to explain random changes. This overconfidence in our abilities hides the real reasons behind many events, making us prone to predictable judgment errors because we do not account for the significant influence of chance.

“The feeling that we are in control is often an illusion. The feeling that we are not in control is often a reality.”

In the book: Mlodinow discusses the hot hand fallacy in basketball, where players and fans believe a player has a 'hot hand' after making several shots, despite statistical analysis showing that sequences of hits and misses are consistent with random chance. He also cites studies on stock market performance, where active fund managers often fail to consistently beat market averages, suggesting that their 'skill' is largely indistinguishable from random fluctuation.

Try this: Cultivate intellectual humility by consciously questioning whether an outcome was due to skill or luck. Before making a decision, consider a counterfactual: what if a different, random variable had occurred? This helps calibrate expectations and reduces overconfidence, leading to more robust strategies that acknowledge uncertainty.

Related ideas: illusion-of-control, cognitive-bias, hot-hand-fallacy

Regression to the Mean

Extreme performances tend to be followed by more average ones.

Regression to the mean is a basic statistical idea often confused with cause-and-effect. Mlodinow explains that when an outcome is affected by both skill and chance, an exceptionally good or bad result is likely due to unusually good or bad luck. Following performances are thus likely to be closer to the average, as extreme luck probably will not repeat. This idea is widely misunderstood, leading to wrong conclusions, such as thinking that criticism improved performance (when it was just returning from an unusually bad period) or that a reward caused a decline (when it followed an unusually good period). Understanding this helps us avoid mistaking random changes for meaningful trends.

“Extremely good or bad performances are often followed by more average ones, not because of some mystical balancing force, but because extreme events are often a combination of skill and extreme luck.”

In the book: Mlodinow uses the example of flight instructor experiments, where instructors believed praise for good landings led to worse subsequent landings, and criticism for bad landings led to better subsequent ones. In reality, exceptionally good or bad landings were simply regressing to the pilot's average performance, making it appear as though the feedback caused the change, rather than it being a statistical inevitability.

Try this: When evaluating performance, especially after an extreme result, remember to account for regression to the mean. Don't attribute an improvement or decline solely to an intervention if the previous result was an outlier. This helps in designing more effective feedback systems and avoiding superstitious conclusions about causality.

Related ideas: regression-to-the-mean, statistical-fallacy, causality-vs-correlation

The Law of Large Numbers (and Small Ones)

Large samples reveal underlying probabilities; small samples can be wildly misleading.

Mlodinow highlights the important difference between large and small sample sizes when looking at data. The Law of Large Numbers states that as the number of trials increases, the observed frequency of an event will approach its true probability. This is why casinos make money over time, even with random results. However, people often fall for the 'Law of Small Numbers,' drawing firm conclusions from too little data. Small samples are very sensitive to random changes, making them unrepresentative and leading to incorrect generalizations. This misunderstanding fuels superstitions, misinterprets medical studies, and distorts public perception, as people readily accept personal stories over solid statistical data.

“The law of large numbers assures us that if we repeat a random experiment a large enough number of times, the average of the results will tend to be close to the expected value. The law of small numbers, however, is a fallacy.”

In the book: Mlodinow discusses how small towns often show higher rates of certain diseases (like cancer clusters) purely by chance, because their smaller populations mean that a few extra cases can dramatically skew the percentage, whereas in larger cities, those same few cases would be diluted. He also mentions the gambler's fallacy, where people believe that after a run of reds on a roulette wheel, black is 'due,' ignoring that each spin is an independent event.

Try this: Always question the sample size when presented with data or anecdotes. Be skeptical of conclusions drawn from small datasets, especially in areas like health, finance, or social trends. Prioritize studies with large, representative samples to gain a more accurate understanding of underlying probabilities.

Related ideas: law-of-large-numbers, gamblers-fallacy, sample-size-bias

The Power of Pathways

The number of ways an event can occur profoundly impacts its probability.

Mlodinow stresses that probability is not just about how likely a single event is, but also about the number of ways that event can happen. This idea, based on combinatorics, explains why some outcomes appear more often than others, even if individual small events are equally likely. For example, there are many more ways to get an average test score than an exceptionally high or low one. Our natural understanding often misses this 'pathway' factor, causing us to misjudge the likelihood of complex events. Understanding this helps explain why 'average' is the most common outcome and why truly unusual events are rare, even if theoretically possible.

“The number of paths to a particular outcome can often be as important as the individual probabilities of each step along the way.”

In the book: Mlodinow illustrates this with the example of a Galton board (or bean machine), where balls drop through a series of pegs. While each individual turn at a peg is random (left or right), the balls accumulate in a bell-shaped curve at the bottom, because there are many more paths that lead to the center bins than to the extreme outer bins. This visually demonstrates how the number of pathways creates the normal distribution.

Try this: When assessing the probability of a complex outcome, don't just consider the likelihood of one specific sequence of events. Instead, think about all the possible combinations and pathways that could lead to that outcome. This perspective helps in understanding why 'average' is so common and 'extreme' is so rare.

Related ideas: combinatorics, normal-distribution, probability-theory

Subjective Probability and Bayesian Thinking

Our beliefs should be updated based on new evidence, not held rigidly.

Mlodinow explores subjective probability, where probabilities reflect how much we believe in an event, rather than an objective frequency. He introduces Bayesian thinking as a way to update these beliefs. Instead of sticking to initial assumptions, Bayesian inference teaches us to adjust our prior probabilities (initial beliefs) with new evidence to form posterior probabilities (updated beliefs). This is a tool for rational decision-making, helping us move past personal stories and confirmation bias. It is especially important in fields like medicine, where understanding the true positive and false positive rates of tests is essential for accurate diagnosis and avoiding unnecessary worry or complacency.

“Probability is not just a property of the world, but also a property of our knowledge about the world.”

In the book: Mlodinow explains the classic medical diagnosis problem: if a disease affects 1% of the population and a test is 90% accurate (90% true positive, 10% false positive), what is the probability that someone who tests positive actually has the disease? Most people overestimate, neglecting the low prior probability of having the disease. Bayesian analysis reveals the actual probability is much lower than intuition suggests, due to the high number of false positives in a low-prevalence population.

Try this: Actively seek out new information that might challenge your existing beliefs. When encountering new data, use it to update, rather than merely confirm, your prior probabilities. This iterative process of belief revision is fundamental to critical thinking and adapting to a changing world.

Related ideas: bayesian-inference, subjective-probability, confirmation-bias, prior-probability

The Fallacy of Determinism

Many outcomes we perceive as predetermined are merely random paths taken.

One of Mlodinow's main points is that our universe, and especially our lives, are less predetermined than we like to think. We tend to create convincing stories after the fact to explain successes and failures, giving random events purpose or inevitability. This backward-looking determinism ignores the many other paths that could have been taken and the significant role of chance. From career success to scientific discoveries, many outcomes that seem to result from genius or careful planning are, in fact, the result of a 'drunkard's walk' – a series of random steps that, in hindsight, can be woven into a coherent story. Recognizing this helps us appreciate how fragile success is and the resilience needed to navigate an unpredictable world.

“We tend to mistake the outcome of a random process for the result of some deep and meaningful plan.”

In the book: Mlodinow discusses the career paths of successful individuals, noting that many attribute their success to a clear vision or specific decisions, while often overlooking the fortunate encounters, unexpected opportunities, or even failures that randomly diverted them onto a successful trajectory. He also alludes to the 'butterfly effect' in chaos theory, where small, random perturbations can lead to vastly different outcomes.

Try this: Resist the urge to create overly simplistic, linear narratives for success or failure. Acknowledge the role of serendipity and random events in your own life and the lives of others. This fosters empathy and a more realistic view of achievement, encouraging adaptability rather than rigid adherence to a 'master plan.'

Related ideas: determinism, serendipity, narrative-fallacy, retrospective-bias

Risk Perception and Decision Making

Our emotional response to risk often overrides rational statistical assessment.

Mlodinow criticizes how our view of risk is often distorted, leading to illogical decisions. We tend to overreact to vivid, emotional, and easily remembered information (availability heuristic) while downplaying more common, less dramatic threats. This explains why people fear flying more than driving, even though driving is statistically much more dangerous. The media often makes this worse by focusing on sensational events, further skewing our risk assessment. Understanding the statistical realities of risk, instead of relying on gut feelings or media portrayals, is important for making informed choices about health, safety, and investments, allowing us to use our attention and resources more effectively.

“The human mind is not built to understand probabilities well, and the media often exploits this weakness.”

In the book: Mlodinow discusses how people often overestimate the likelihood of rare, dramatic events (like plane crashes or terrorist attacks) due to their vivid portrayal in the media, while underestimating the probability of more common, mundane risks (like car accidents or lifestyle-related diseases). He also touches upon the anchoring effect, where an initial piece of information biases subsequent judgments.

Try this: When evaluating risks, actively seek out objective statistical data rather than relying on anecdotes, media headlines, or emotional responses. Compare risks in a broader context and challenge your initial gut reactions to ensure decisions are based on actual probabilities, not just perceived threats.

Related ideas: risk-perception, availability-heuristic, emotional-bias, anchoring-effect

The Drunkard's Walk of Molecules and Markets

Random walks underpin diverse phenomena, from physics to finance.

At the core of Mlodinow's book is the 'drunkard's walk' or random walk, a mathematical model where each step is taken in a random direction. He shows how this simple model, originally used to describe the erratic movement of molecules (Brownian motion), applies to many complex systems. From stock market changes and disease spread to individual life paths, many things that seem to have patterns are, in fact, best described as random walks. Recognizing this helps us understand the inherent unpredictability in many systems and warns against over-interpreting short-term trends as signs of long-term direction or skill.

“The random walk is a mathematical model that describes a path consisting of a sequence of random steps.”

In the book: Mlodinow explains Brownian motion, where pollen grains suspended in water move randomly due to collisions with water molecules. He then extends this to financial markets, arguing that stock prices often follow a random walk, meaning past price movements are not reliable predictors of future movements, making consistent stock picking extremely difficult.

Try this: When observing volatile systems like financial markets or even career trajectories, be wary of attributing every fluctuation to a deliberate cause. Understand that inherent randomness can drive significant movement. This encourages a more patient, less reactive approach, focusing on long-term probabilities rather than short-term noise.

Related ideas: random-walk, brownian-motion, stochastic-processes, market-efficiency

The Tyranny of Small Differences

Minute random advantages can compound into massive disparities over time.

Mlodinow reveals how seemingly small, random advantages or disadvantages can, over time, grow into large differences in outcomes. This 'tyranny of small differences' explains why, for example, a slightly better student early on might get access to better resources, which further improves their performance, leading to a big gap later. It is not always about a single, decisive moment of genius or failure, but often a chain of small, random benefits or setbacks accumulating. This idea challenges meritocratic ideals, suggesting that while effort matters, initial luck and subsequent lucky breaks play a bigger role in shaping destinies than we usually admit.

“In a world ruled by chance, the initial conditions, even tiny random differences, can have enormous consequences over time.”

In the book: Mlodinow cites studies on the careers of scientists, athletes, and artists, showing how early, often random, recognition or opportunity can create a positive feedback loop, leading to disproportionate success. He might also allude to the Matthew Effect, where 'the rich get richer and the poor get poorer,' demonstrating how initial advantages snowball.

Try this: Recognize that 'success' is often a complex interplay of effort, talent, and accumulated random advantages. Advocate for systems that mitigate the tyranny of small differences by providing more equitable starting points and opportunities for those who may have experienced early random disadvantages. Practice humility in your own successes and empathy for others' struggles.

Related ideas: compounding-effects, initial-conditions, inequality, matthew-effect

Embracing Uncertainty for Better Decisions

True wisdom lies in understanding and adapting to a random world, not fighting it.

Ultimately, Mlodinow's work encourages us to accept uncertainty rather than deny it. Our deep need for meaning and control often leads us to create elaborate, but false, explanations for random events. By understanding the mathematical principles of randomness—from regression to the mean to the law of large numbers—we can develop a more realistic and balanced view of the world. This does not mean giving in to fatalism, but rather giving ourselves the mental tools to make better decisions, manage expectations, and build resilience in the face of unpredictable outcomes. True wisdom, Mlodinow suggests, comes from recognizing where randomness rules and where our efforts truly make a difference.

“The greatest obstacle to understanding the role of randomness in our lives is our human need for meaning and control.”

In the book: Throughout the book, Mlodinow consistently contrasts intuitive, often flawed, human reasoning with rigorous statistical analysis. He shows how acknowledging randomness in sports, finance, and everyday life can lead to more effective strategies, such as focusing on process over outcome, or diversifying investments rather than chasing 'hot' stocks.

Try this: Integrate an understanding of randomness into your daily decision-making. Focus on processes that are robust to random fluctuations, rather than trying to predict or control every outcome. Develop a mindset that accepts uncertainty as an inherent part of life, fostering adaptability and reducing anxiety about factors beyond your control.

Related ideas: uncertainty-acceptance, decision-making, resilience, statistical-literacy

The Drunkard's Walk Quotes

“The mind is a wonderful thing. It starts working the minute you're born and never stops working until you get into a classroom and are asked to solve a problem.”
Mlodinow's humorous observation on learning and problem-solving.
“We often confuse the fact that the world is deterministic with the idea that we can predict it.”
Discussing the difference between determinism and predictability, especially in complex systems.
“The laws of probability, like the laws of physics, are not suggestions. They are laws.”
Emphasizing the fundamental and unyielding nature of probability laws.
“Chance is a more fundamental concept than determinism.”
A provocative statement challenging the intuitive dominance of deterministic thinking.
“The human mind is a story generator, not a logic machine.”
Explaining why humans often prefer narratives and causal explanations over statistical reasoning.
“Sometimes a truly random event just looks like it was caused by something.”
Highlighting the human tendency to seek patterns and causes even in pure randomness.
“The world is not as predictable as we think it is, nor as unpredictable as we sometimes fear.”
A nuanced perspective on the balance between order and chaos in the universe.
“We are much more likely to remember when our prediction was right than when it was wrong.”
Discussing confirmation bias and selective memory in evaluating our foresight.
“Luck is not a magical force, but a statistical reality.”
Demystifying the concept of luck by grounding it in the principles of probability.
“Correlation is not causation, but it sure is a hint.”
A common statistical adage, acknowledging the practical utility of correlation while warning against overinterpretation.
“The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge.”
A powerful statement about the danger of overconfidence and unexamined beliefs.
“The more unlikely an event, the more likely we are to search for a cause.”
Explaining the psychological drive to find explanations for rare or surprising occurrences.
“Our brains are wired to see patterns, and sometimes those patterns are just noise.”
Discussing the evolutionary advantage and potential pitfalls of human pattern recognition.
“Understanding probability means understanding the world better, and understanding the world better means making better decisions.”
Summarizing the core message and practical benefit of the book.

Quiz

10 questions on the main ideas. Each answer comes with a short explanation.

  1. Question 1

    According to Mlodinow, what is one of the primary reasons humans struggle to understand randomness?

    1. Lack of formal education in probability theory.
    2. The inherent complexity of random processes.
    3. Psychological illusions that prevent accurate understanding.
    4. The fact that randomness is entirely unpredictable.
    Show answer

    C. Psychological illusions that prevent accurate understanding. Mlodinow emphasizes that psychological illusions often prevent us from accurately interpreting random events, leading to misjudgments in various aspects of life.

  2. Question 2

    The book suggests that the same mathematical tools used to understand the random paths of molecules can be applied to what everyday phenomena?

    1. Quantum physics and black holes.
    2. Winning the lottery, road safety, and success of film stars.
    3. The movement of planets and celestial bodies.
    4. The growth patterns of plants and animals.
    Show answer

    B. Winning the lottery, road safety, and success of film stars. Mlodinow argues that the principles of randomness are universally applicable, from the microscopic world to seemingly complex human endeavors like success in entertainment or predicting lottery outcomes.

  3. Question 3

    Which author provided a notable endorsement for 'The Drunkard's Walk', calling it 'a wonderfully readable guide to how the mathematical laws of randomness affect our lives'?

    1. Carl Sagan
    2. Richard Dawkins
    3. Stephen Hawking
    4. Neil deGrasse Tyson
    Show answer

    C. Stephen Hawking The book's description explicitly quotes Stephen Hawking's endorsement, highlighting the book's accessibility and relevance.

  4. Question 4

    Mlodinow often uses examples from sports to illustrate the role of randomness. What is a common pitfall in interpreting athletic performance that the book likely addresses?

    1. Overemphasizing the role of coaching in team victories.
    2. Attributing all success solely to innate talent, ignoring practice.
    3. Mistaking streaks of good or bad luck for underlying skill changes.
    4. Underestimating the physical conditioning required for elite sports.
    Show answer

    C. Mistaking streaks of good or bad luck for underlying skill changes. The book frequently discusses how 'hot hands' or 'cold streaks' in sports are often misattributed to changes in skill rather than the natural fluctuations of random processes.

  5. Question 5

    When discussing 'making sense of a blood test', Mlodinow likely emphasizes the importance of understanding what aspect of medical results?

    1. The absolute numerical value of a single test result.
    2. The doctor's subjective interpretation of the patient's symptoms.
    3. The statistical probability of false positives or negatives and the context of the reference range.
    4. The cost-effectiveness of different diagnostic procedures.
    Show answer

    C. The statistical probability of false positives or negatives and the context of the reference range. Mlodinow would likely highlight how medical tests, despite appearing definitive, are subject to probabilities, measurement errors, and the base rate fallacy, requiring a nuanced statistical understanding.

  6. Question 6

    The title 'The Drunkard's Walk' is a metaphor for what concept explored in the book?

    1. The unpredictable and undirected path of random events.
    2. The human tendency to make irrational decisions under influence.
    3. The journey of scientific discovery, often fraught with errors.
    4. The social and economic impact of alcohol consumption.
    Show answer

    A. The unpredictable and undirected path of random events. The 'drunkard's walk' or 'random walk' is a classic statistical model representing a path consisting of a sequence of random steps, perfectly illustrating the book's theme of how randomness shapes our lives.

  7. Question 7

    When Mlodinow discusses 'stock-picking', he likely argues against which common belief?

    1. That individual stocks are inherently riskier than mutual funds.
    2. That a truly skilled investor can consistently outperform the market through insight alone.
    3. That diversified portfolios are less profitable than concentrated ones.
    4. That government regulation has no impact on stock market volatility.
    Show answer

    B. That a truly skilled investor can consistently outperform the market through insight alone. A core argument in the book is that much perceived skill in fields like stock-picking is often attributable to random fluctuations, and consistently beating the market is extremely difficult due to the nature of chance.

  8. Question 8

    The book aims to reveal the 'truth about the success of sporting heroes and film stars'. What 'truth' does it primarily unveil regarding their success?

    1. Their success is solely a result of relentless hard work and dedication.
    2. Their success is largely due to superior genetic predispositions for their respective fields.
    3. While skill is necessary, a significant component of their extraordinary success is often due to chance and lucky breaks.
    4. Their success is primarily manufactured by powerful media and marketing campaigns.
    Show answer

    C. While skill is necessary, a significant component of their extraordinary success is often due to chance and lucky breaks. Mlodinow argues that while talent and effort are crucial, the extreme success of individuals in highly competitive fields often involves a substantial element of luck and being in the right place at the right time.

  9. Question 9

    Mlodinow would likely use the concept of 'regression to the mean' to explain which phenomenon?

    1. Why children of exceptionally tall parents are often shorter than their parents.
    2. Why certain species are more likely to survive environmental changes.
    3. Why market bubbles eventually burst after reaching extreme highs.
    4. Why complex systems tend to revert to simpler states over time.
    Show answer

    A. Why children of exceptionally tall parents are often shorter than their parents. Regression to the mean is a statistical phenomenon where extreme values tend to be followed by values closer to the average. This is famously illustrated with examples like the heights of children compared to their parents, or the performance of athletes after an exceptionally good or bad game.

  10. Question 10

    The book warns against becoming 'another victim of chance' by implying that an understanding of randomness can help us to:

    1. Eliminate all uncertainty from our lives and make perfect predictions.
    2. Better assess risks, make more rational decisions, and avoid misinterpreting random events as deterministic.
    3. Manipulate random outcomes to our advantage, such as winning lotteries.
    4. Accept all outcomes as inevitable and stop trying to influence events.
    Show answer

    B. Better assess risks, make more rational decisions, and avoid misinterpreting random events as deterministic. The core message is not to eliminate randomness, but to understand its pervasive influence, allowing for more informed decision-making, better risk assessment, and avoiding common cognitive biases that lead to misinterpretations.

The Drunkard's Walk FAQ

What is 'The Drunkard's Walk' about?

'The Drunkard's Walk' by Leonard Mlodinow explores how randomness and uncertainty profoundly influence our lives, yet we often fail to grasp their significance. The book uses mathematical principles to reveal the hidden role of chance in everything from lotteries and sports to stock markets and medical diagnoses.

Does understanding randomness help in making better decisions?

Yes, the book argues that recognizing the pervasive nature of randomness can help individuals overcome psychological biases and make more informed decisions. By understanding the true impact of chance, one can avoid misinterpreting outcomes and attributing them solely to skill or intention.

What is the 'drunkard's walk' concept?

The 'drunkard's walk' is a metaphor for a random walk, where a path consists of a succession of random steps. Mlodinow uses this concept to illustrate how seemingly purposeful or successful outcomes can often be the result of a series of random events, much like a drunkard's unpredictable path.

How does 'The Drunkard's Walk' differ from other books on probability or statistics?

While other books may focus on the technical aspects of probability, Mlodinow's work emphasizes the psychological illusions that prevent us from understanding randomness in daily life. It bridges the gap between mathematical theory and its practical implications for human behavior and decision-making, often using engaging anecdotes and real-world examples.

Can 'The Drunkard's Walk' help me win the lottery or predict stock prices?

No, the book does not offer strategies for predicting or manipulating random events like lottery wins or stock prices. Instead, it aims to help readers understand why such events are inherently unpredictable and how to avoid common fallacies that lead people to believe they can beat the odds.

What kind of examples does Mlodinow use to illustrate randomness?

Mlodinow draws on a wide range of examples, including the success of athletes and movie stars, the outcomes of medical tests, the fluctuations in the stock market, and even the paths of molecules. These diverse scenarios help demonstrate how randomness underpins many aspects of our seemingly ordered world.

Is 'The Drunkard's Walk' suitable for someone without a strong math background?

Absolutely. Mlodinow makes complex mathematical concepts accessible and engaging for a general audience. While it deals with probability, the book focuses on intuitive understanding and real-world applications rather than dense equations, making it enjoyable for readers of all backgrounds.

How does the book address the role of skill versus luck in success?

The book meticulously explores how often what we perceive as skill or genius can, in fact, be significantly influenced by luck or random chance. Mlodinow provides frameworks for discerning the contributions of both skill and randomness, particularly in fields like sports, business, and entertainment.

About the author

Leonard Mlodinow

Leonard Mlodinow

Leonard Mlodinow is an American theoretical physicist and mathematician, screenwriter and author. In physics, he is known for his work on the large N expansion, a method of approximating the spectrum of atoms based on the consideration of an infinite-dimensional version of the problem, and for his work on the quantum theory of light inside dielectrics.

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