I usually try to control myself and resist giving compliments or unsolicited advice to people I don’t know well. But I recently did just that. I emailed letters to Lee Scott, C.E.O. of Wal-Mart, and Jeff Immelt, C.E.O. of General Electric. I wanted to share with them some of my thoughts about their businesses. Both companies are leviathans that have struggled recently to grow in ways that satisfy shareholders. Wal-Mart, I wrote to Scott, has created a phenomenal opportunity for itself by offering in-store health clinics and, in the process, may transform the U.S. health-care industry. G.E., I told Immelt, could boost its fortunes by complementing its experience in power generation with building an electric car.

In my position as a lecturer at Stanford Graduate School of Business, I’ve been working with my colleague Professor Robert Burgelman to examine how large companies can defeat the law of big numbers. Successful businesses sooner or later encounter a situation in which the reward for their success becomes a punishment of sorts. The reward is that they get big. The punishment is that when they get big, it gets harder and harder for them to grow. And then their investors pile on the abuse.

In looking at various companies that have been hindered by their own success, we found that under certain conditions a firm can create a new growth spurt for itself by entering an entirely different industry. The target industry must be stagnant and populated with companies that cling to doing business the way they always have. The corporation that enters this environment with an innovative product or service can shake up the status quo and reap big profits. Burgelman and I call this phenomenon cross-boundary disruption, or XBD for short.

The defining example of this kind of move is Apple’s incursion into the sluggish music business with the introduction of the iPod in 2001 and then the iTunes music store in 2003. At the time, Apple faced market saturation in its niche. (Its relatively high-end computers were stuck with a single-digit market share.) It had all the resources of an established, well-run corporation: highly skilled employees, brand appeal, and access to capital. And it was hungry for growth. Since Apple entered the music business, the company’s profit has increased more than 3,000 percent, from $57 million in 2003 to nearly $2 billion in 2006.

The XBD phenomenon is something separate from the more familiar pattern of startups forging industry change in steps. Clayton Christensen described that process in his book The Innovator's Dilemma. In Christensen’s scenario, a small company penetrates an industry by first establishing a position in the least demanding portion of it and then progressing into more-demanding segments. Christensen shows how minimills entered the U.S. steel industry in the 1970s by concentrating on the low-margin area of the market that the established players had largely given up on. The new companies grew until they were strong enough to attack larger and larger market segments. By 2000, these minimills had increased their production to almost 50 percent of the raw-steel market.

Cross-boundary disruption is different. I’m talking about established giants seeking to transform markets other than their own. It's Apple jumping into music. It’s Wal-Mart entering health care. Or as my email to Immelt urged, it could be G.E. building an electric car and taking on the energy industry. These are companies big and powerful enough to solve intractable, industrywide problems and produce lasting change.